Induced high order aberrations corresponding to geometrical transformations

ABSTRACT

Wavefront measurements of eyes are typically taken when the pupil is in a first configuration in an evaluation context. The results can be represented by a set of basis function coefficients. Prescriptive treatments are often applied in a treatment context, which is different from the evaluation context. Hence, the patient pupil can be in a different, second configuration, during treatment. Systems and methods are provided for determining a transformed set of basis function coefficients, based on a difference between the first and second configurations, which can be used to establish the vision treatment.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. patent applicationSer. No. 12/725,575, filed Mar. 17, 2010, which is a continuation ofU.S. patent application Ser. No. 11/832,408, filed Aug. 1, 2007, nowU.S. Pat. No. 7,695,136. This application is also related to U.S. patentapplication Ser. No. 12/722,881, filed Mar. 12, 2010, which is acontinuation of U.S. patent application Ser. No. 11/676,094, filed Feb.16, 2007, which claims the benefit of U.S. Provisional Application No.60/776,289, filed Feb. 24, 2006. The content of each of the above listedapplications is incorporated herein by reference.

BACKGROUND OF THE INVENTION

Embodiments of the present invention relate to systems and methods fortreating vision in a patient. Particular embodiments encompass treatmenttechniques that account for geometrical transformations in wavefrontmaps.

Ocular aberrations of human eyes can be measured objectively withwavefront technology. In the various fields of optics, wavefrontaberrations have traditionally been represented by Zernike polynomials.Wavefront measurements of eyes are normally taken when the pupil isrelatively large, and the results are often represented by a set ofZernike coefficients. Different sets of Zernike coefficients can becalculated to represent aberrations at smaller pupil sizes. Pupil sizesmay change according to the lighting environment or context where apatient is situated, for example. Recently described techniques allowscaling of the expansion coefficients with Zernike polynomials, yetthere remains a need for scaling approaches that can used with otherbasis functions, such as Taylor monomials. Moreover, there remains aneed for techniques that account for other geometrical transformations,such as pupil center shift and cyclorotation.

Many current approaches to wavefront refraction and ablation shapedesign, such as for the treatment of presbyopia, do not consider thecontribution of induced high order aberrations due to geometricaltransformations which may include pupil constriction, pupil centershift, or cyclorotation. Accordingly, there is a need for a generalanalytical errorless approach for determining a new set of coefficientsof any basis functions from an original set when an ocular wavefront mapevokes a geometrical transformation that includes pupil constriction,cyclorotation, or pupil center shift, or any combination thereof. Thereis also a need for a general geometrical transformation technique thatdoes not have the restriction of a sub-area definition after such ageometrical transformation. Relatedly, there is a need for an optimalanalytical errorless approach for calculating wavefront refractions whena geometrical transformation occurs. There is also a need for tissueablation profiles that include the adjustment of such geometricaltransformations for the correction of high order aberrations. Further,there is often an error or discrepancy between the manifest refractionand wavefront refraction. There is a need for systems and methods forcombining a CustomVue treatment with a shifted presbyopic treatment.Embodiments of the present invention provide solutions for visiontreatment that address at least some of these needs.

SUMMARY OF THE INVENTION

Ocular wavefront maps typically change when the pupil parameters change.These map changes can reflect geometrical transformations such as pupilconstrictions, cyclorotations, and pupil center shifts. Any one of thesegeometrical transformations, or any combination thereof, can result in adifferent set of Zernike or other basis function coefficients, thusaffecting the calculation of wavefront refractions and the design ofvision correction profiles such as for the correction or treatment ofpresbyopia. Advantageously, embodiments of the present invention providesystems and methods for calculating wavefront refractions and fordesigning optimal or optimized ablation shapes for vision correctionwhen a geometrical transformation occurs in an ocular wavefront map.Often these techniques involve improvements in accuracy for wavefrontdeterminations. Embodiments disclosed herein are well suited for use inmany vision correction and treatment modalities, including withoutlimitation corneal ablation, contact lenses, intraocular lenses, andspectacle lenses.

Hence, an exemplary treatment method may include obtaining a wavefrontof the patient's eye when the patient is in an evaluation environment orcontext and the eye is in a certain geometrical configuration. Thewavefront can be characterized by a set of coefficients for a basisfunction. The method can also include exposing the patient to atreatment environment or context, such that the eye is in a newgeometrical configuration. A new wavefront can be determined based onthe earlier geometrical configuration of the eye, the original set ofcoefficients, and the new geometrical configuration. The new wavefrontmap can be characterized by a new set of coefficients for the basisfunction. Based on the new wavefront, it is possible to establish aprescription for the patient. The method can also include treating thepatient with the prescription.

Embodiments of the present invention provide systems and methods forcalculating a new set of coefficients of any basis functions from anoriginal set when an ocular wavefront map evokes a geometricaltransformation that includes pupil constrictions, cyclorotation, orpupil center shift, based on a general analytical or errorless approach.For example, in the case of a basis function representation of one ormore particular ocular aberrations, embodiments disclosed herein providetechniques for determining a new set of basis function coefficients thatreflect changes in pupil parameters or geometrical transformations.These techniques can be used to determine or characterize howgeometrical transformations affect visual performance and the refractiondeterminations. It has been discovered that any basis function which canbe separated into radial polynomials and a triangular function can becharacterized by a generic pupil rescaling formula (GPRF).

Embodiments also provide a general geometrical transformation approachthat does not have the restriction of a sub-area definition after such ageometrical transformation.

Embodiments encompass cases where the set of basis functions is the setof Taylor monomials, or Zernike polynomials. Embodiments also encompasscases where the geometrical transformation includes only a pupilconstriction, only a cyclorotation, only a pupil center shift, or acombination of any two of these geometrical transformations, or acombination of all three. Embodiments of the present invention alsoprovide systems and methods for calculating wavefront refractions when ageometrical transformation occurs, based on an optimal analytical orerrorless approach for calculating. Embodiments also provide techniquesfor establishing tissue ablation profiles and other vision treatmentprofiles that include adjustments for, or otherwise consider, suchgeometrical transformations for the correction of high orderaberrations. In some embodiments, systems and methods provide for thepresbyopic treatments where the presbyopic shape is decentered, rotated,or otherwise shifted, and the profile is combined with the customizedtreatment, such as a CustomVue treatment. Embodiments also providetreatments that correct for or address errors or discrepancies betweenthe manifest refraction and wavefront refraction.

As noted above, embodiments of the present invention provide techniquesfor scaling several types of basis functions. Moreover, embodimentsprovide techniques for obtaining new coefficients due to pupil parameterchanges such as pupil constriction, decentration, and cyclorotation.Techniques can include any desired combination, in any desired order.Pupil decentration embodiments may involve x- and y-shifts.Cyclorotation embodiments may involve the angle of rotation. In somecases, it is assumed that the ocular aberrations are invariant of thecoordinate change, and the aberrations are manifested from the optics ofthe eye, such as the cornea, the crystalline lens, and the mediatherebetween. The relative position and the property of opticalcomponents often does not change because of the pupil constriction,decentration, and cyclo-rotation. Hence, it is possible to establish anoriginal domain that defines the original ocular aberrations, or in abroad sense, an original function. When the domain changes, it ispossible to establish a new domain within the original domain. When thenew domain is determined, various approaches can be used to fit theocular aberration, or in a broad sense, a function, with a complete setof basis functions, such as Zernike polynomials, Fourier series, Taylormonomials, and the like. This approach can be applied to pupil parameterchanges or geometrical transformations such as pupil constriction,decentration, and cyclorotation.

When a wavefront map is captured, it may be desirable to design anablation treatment based on adjustments to the size or orientation ofthe map. Wavefront exams can be processed to adjust for changes in pupilsize or alignment. For example, a treatment area may not be exactly thesame as the area under which a wavefront is captured. Hence, it can beuseful, after determining an original set of basis functioncoefficients, to determine a new set of basis function coefficientscorresponding to a different ocular configuration.

In a first aspect, embodiments of the present invention encompass asystem for establishing a prescription that mitigates or treats a visioncondition of an eye in a particular patient. The system can include, forexample, a first module having a tangible medium embodyingmachine-readable code that accepts a first geometrical configuration ofthe eye, a second module having a tangible medium embodyingmachine-readable code that determines an original set of coefficientsfor a basis function characterizing the first geometrical configuration.The basis function can be separated into a product of a first set ofradial polynomials and a first triangular function. The system can alsoinclude a third module having a tangible medium embodyingmachine-readable code that accepts a second geometrical configuration ofthe eye, and a fourth module having a tangible medium embodyingmachine-readable code that determines a transformed set of coefficientsfor the basis function. The transformed set of coefficients can be basedon the first geometrical configuration of the eye, the original set ofcoefficients, and the second geometrical configuration of the eye. Thesystem can also include a fifth module having a tangible mediumembodying machine-readable code that derives the prescription for theparticular patient based on the transformed set of coefficients. Theprescription may mitigate or treat the vision condition of the eye. Insome cases, a difference between the first geometrical configuration ofthe eye and the second geometrical configuration of the eye ischaracterized by a pupil center shift. In some cases, a differencebetween the first geometrical configuration of the eye and the secondgeometrical configuration of the eye is characterized by acyclorotation. In some cases, a difference between the first geometricalconfiguration of the eye and the second geometrical configuration of theeye is characterized by a pupil constriction. A basis function mayinclude a Zernike basis function, a Taylor basis function, a Seidelbasis function, or the like.

In another aspect, embodiments of the present invention provide methodsfor establishing a prescription that mitigates or treats a visioncondition of an eye in a particular patient. An exemplary methodincludes inputting a first geometrical configuration of the eye, anddetermining an original set of coefficients for a basis functioncharacterizing the first geometrical configuration, where the basisfunction can be separated into a product of a first set of radialpolynomials and a first triangular function. The method can also includeinputting a second geometrical configuration of the eye, and determininga transformed set of coefficients for the basis function, where thetransformed set of coefficients are based on the first geometricalconfiguration of the eye, the original set of coefficients, and thesecond geometrical configuration of the eye. The method can also includeestablishing the prescription for the particular patient based on thetransformed set of coefficients, where the prescription mitigates ortreats the vision condition of the eye. In some cases, a differencebetween the first geometrical configuration of the eye and the secondgeometrical configuration of the eye is characterized by a pupil centershift. In some cases, a difference between the first geometricalconfiguration of the eye and the second geometrical configuration of theeye is characterized by a cyclorotation. In some cases, a differencebetween the first geometrical configuration of the eye and the secondgeometrical configuration of the eye is characterized by a pupilconstriction. A basis function may include a Zernike basis function, aTaylor basis function, a Seidel basis function, or the like.

In another aspect, embodiments of the present invention encompassmethods for treating a particular patient with a prescription thatmitigates or treats a vision condition of an eye of the patient. Forexample, a method can include obtaining a first wavefront map of the eyethat corresponds to a first geometrical configuration of the eye in anevaluation context, where the first wavefront map is characterized by anoriginal set of coefficients for a basis function that can be separatedinto a product of a first set of radial polynomials and a firsttriangular function. The method can also include determining a secondwavefront map of the eye that corresponds to a second geometricalconfiguration of the eye in a treatment context, where the secondwavefront map is characterized by a transformed set of coefficients forthe basis function that is based on the first geometrical configurationof the eye, the original set of coefficients, and the second geometricalconfiguration of the eye. Further, the method can include establishingthe prescription for the particular patient based on the transformed setof coefficients, and treating the patient with the prescription tomitigate or treat the vision condition of the eye. In some cases, adifference between the first geometrical configuration of the eye andthe second geometrical configuration of the eye is characterized by apupil center shift. In some cases, a difference between the firstgeometrical configuration of the eye and the second geometricalconfiguration of the eye is characterized by a cyclorotation. In somecases, a difference between the first geometrical configuration of theeye and the second geometrical configuration of the eye is characterizedby a pupil constriction. A basis function may include a Zernike basisfunction, a Taylor basis function, a Seidel basis function, or the like.

In still another aspect, embodiments of the present invention encompasssystems and methods for determining a high order aberration induced by achange in geometrical configuration in an eye of a patient. Exemplarymethods may include inputting a first geometrical configuration of theeye, and inputting an original set of coefficients for a basis functioncharacterizing the first geometrical configuration of the eye. In somecases, the basis function can be separated into a product of a first setof radial polynomials and a first triangular function. Methods may alsoinclude inputting a second geometrical configuration of the eye, andinputting a transformed set of coefficients for the basis function. Insome cases, the transformed set of coefficients are based on the firstgeometrical configuration of the eye, the original set of coefficients,and the second geometrical configuration of the eye. Methods may alsoinclude determining the induced high order aberration based on thetransformed set of coefficients. According to some embodiments, adifference between the first geometrical configuration and the secondgeometrical configuration is related to or corresponds to a pupil centershift. According to some embodiments, a difference between the firstgeometrical configuration and the second geometrical configuration isrelated to or corresponds to a pupil constriction or a pupil dilation.According to some embodiments, a difference between the firstgeometrical configuration and the second geometrical configuration isrelated to or corresponds to a cyclorotation and a pupil center shift, apupil constriction, or a pupil dilation. In some instances, the basisfunction includes a Zernike basis function. In some instances, the basisfunction includes a Taylor basis function. In some instances, the basisfunction includes a Seidel basis function. The induced high orderaberration can include, for example, coma, secondary coma, trefoil,primary spherical aberration, secondary spherical aberration, secondaryastigmatism, or tertiary astigmatism. Methods may also includedetermining a predicted vision symptom based on the induced high orderaberration. In some cases, the vision symptom comprises a predictednight vision symptom. Optionally, methods may include determining atreatment based on the induced high order aberration. Methods may alsoinclude displaying the transformed set of coefficients for the basisfunction. In some cases, methods may include displaying the induced highorder aberration.

In a further aspect, embodiments of the present invention encompasssystems and methods for determining a high order aberration induced by achange in geometrical configuration in an eye of a patient. Exemplarysystems may include a first module that accepts a first geometricalconfiguration of the eye, and a second module that accepts an originalset of coefficients for a basis function characterizing the firstgeometrical configuration of the eye. In some cases, the basis functioncan be separated into a product of a first set of radial polynomials anda first triangular function. Systems may also include a third modulethat accepts a second geometrical configuration of the eye, and a fourthmodule that determines a transformed set of coefficients for the basisfunction. In some cases, the transformed set of coefficients are basedon the first geometrical configuration of the eye, the original set ofcoefficients, and the second geometrical configuration of the eye.Systems may also include a fifth module that determines the induced highorder aberration based on the transformed set of coefficients.

According to some system embodiments, a difference between the firstgeometrical configuration and the second geometrical configurationcorresponds to a pupil center shift. According to some systemembodiments, a difference between the first geometrical configurationand the second geometrical configuration corresponds to a pupilconstriction or a pupil dilation. According to some system embodiments,a difference between the first geometrical configuration and the secondgeometrical configuration corresponds to a cyclorotation and a pupilcenter shift, a pupil constriction, or a pupil dilation.

In some aspects, embodiments of the present invention encompass computerprogram products for determining a high order aberration induced by achange in geometrical configuration in an eye of a patient. Exemplarycomputer program products may include code for accepting a firstgeometrical configuration of the eye, and code for determining anoriginal set of coefficients for a basis function characterizing thefirst geometrical configuration. In some cases, the basis function canbe separated into a product of a first set of radial polynomials and afirst triangular function. Computer program products may also includecode for accepting a second geometrical configuration of the eye, andcode for determining a transformed set of coefficients for the basisfunction. In some cases, the transformed set of coefficients are basedon the first geometrical configuration of the eye, the original set ofcoefficients, and the second geometrical configuration of the eye.Further, computer program products may include code for determining theinduced high order aberration based on the transformed set ofcoefficients. In some instances, computer program products may includecode for determining a treatment based on the induced high orderaberration. In some instances, computer program products may includecode for displaying the transformed set of coefficients for the basisfunction.

For a fuller understanding of the nature and advantages of the presentinvention, reference should be had to the ensuing detailed descriptiontaken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a laser ablation system according to embodiments ofthe present invention.

FIG. 2 illustrates a simplified computer system according to embodimentsof the present invention.

FIG. 3 illustrates a wavefront measurement system according toembodiments of the present invention.

FIG. 3A illustrates another wavefront measurement system according toembodiments of the present invention.

FIG. 4 shows an illustration of the human eye according to embodimentsof the present invention.

FIGS. 5A and 5B show exemplary illustrations of a human eye in variouscontexts, according to embodiments of the present invention.

FIG. 6 shows an illustration of the human eye according to embodimentsof the present invention.

FIGS. 7A and 7B show exemplary illustrations of wavefront map contourplots for a human eye, according to embodiments of the presentinvention.

FIGS. 8A and 8B show exemplary illustrations of wavefront maps for ahuman eye, according to embodiments of the present invention.

FIGS. 9A and 9B show exemplary illustrations of wavefront maps for ahuman eye, according to embodiments of the present invention.

FIG. 10 shows a graph of effective power curves for sphere and cylinderas a function of pupil size, according to embodiments of the presentinvention.

FIGS. 11A and 11B show illustrations of wavefront maps for a human eye,according to embodiments of the present invention.

FIGS. 12A to 12C show degrees of freedom for rotational eye movements,according to embodiments of the present invention.

FIG. 13 depicts coordinates before and after a cyclorotation, accordingto embodiments of the present invention.

FIGS. 14A to 14D show illustrations of wavefront maps for a human eye,according to embodiments of the present invention.

FIGS. 15A to 15H show illustrations of wavefront contour maps for ahuman eye, according to embodiments of the present invention.

FIGS. 16A to 16G show illustrations of point spread functions andcorresponding simulated images, according to embodiments of the presentinvention.

FIGS. 17A to 17C show illustrations of wavefront maps for a human eye,according to embodiments of the present invention.

FIGS. 18A to 18C show illustrations of wavefront maps for a human eye,according to embodiments of the present invention.

FIGS. 19A to 19H show illustrations of wavefront contour maps for ahuman eye, according to embodiments of the present invention.

FIGS. 20A to 20G show illustrations of point spread functions andcorresponding simulated images, according to embodiments of the presentinvention.

FIGS. 21A and 21B show illustrations of wavefront contour maps for ahuman eye, according to embodiments of the present invention.

FIGS. 22A to 22D show illustrations of wavefront contour maps for ahuman eye, according to embodiments of the present invention.

FIG. 23 shows a graph of curves for primary, secondary, and tertiaryspherical aberration (SA) as a function of pupil size, according toembodiments of the present invention.

FIG. 24 illustrates aspects of the use of diagnostic display and the useof retreatment and enhancement treatment by pupil rescaling techniquesaccording to embodiments of the present invention.

FIG. 25 illustrates aspects of diagnostic display techniques forcomparing or differencing among different exams from the same eye atdifferent timestamps or for different eyes according to embodiments ofthe present invention.

FIG. 26 illustrates aspects of the use of diagnostic display and the useof retreatment and enhancement treatment by geometrical transformationtechniques according to embodiments of the present invention.

FIG. 27 shows aspects of diagnostic display techniques for comparing ordifferencing among different exams from the same eye at differenttimestamps or for different eyes according to embodiments of the presentinvention.

FIG. 28 illustrates aspects of techniques for determining a high orderaberration induced by a change in geometrical configuration in an eye ofa patient according to embodiments of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

Embodiments of the present invention encompass techniques for wavefronttransformation and iris registration, wavefront representation for pupilresizing, wavefront representation for cyclorotation, and wavefrontrepresentation for decentration. Related examples and derivations arealso provided. Pupil resizing approaches encompass Taylor resizingmonomials, Zernike resizing polynomials, and pupil resizing with Seidelseries. Pupil resizing techniques can also involve effective power andthe treatment of presbyopia. Cyclorotation approaches encompasswavefront rotation with Taylor monomials and Zernike polynomials.Decentration approaches encompass wavefront extrapolation, wavefrontdecentration with Taylor monomials and Zernike polynomials, andwavefront refraction of decentered aberrations. Wavefront representationtechniques can also involve wavefront transformation and subsequentrefraction. Embodiments disclosed herein provide refraction calculationformulas for various cases of geometrical transformations, includingrotation, decentration, and constriction.

Embodiments of the present invention provide techniques for rescalingpolynomials that correspond to a patient pupil, and encompass methodsand systems for calculating Zernike resizing polynomials and forderiving scaling for Taylor, Seidel, and other basis functions. In somecases, the present techniques involve a nonrecursive approach. In somecases, the present techniques involve an analytical based approach fordetermining ocular aberrations and refractive treatments. Embodimentsdisclosed herein provide techniques for establishing pupil resizingpolynomials for various basis functions. For example, in the situationwhere an eye presents a wavefront and the pupil of the eye constricts,it is possible to define a circle with a radius that corresponds to theconstricted pupil, and to define an aberration pattern that correspondsto the circle. In some cases, the normalized radius is constricted orcontracted. An epsilon as a ratio can be established that represents aratio of the new smaller pupil radius to the original pupil radius. Inthe original wavefront, the part that is within the boundary when thenormalized pupil is epsilon describes the wavefront that is to beconstricted. It is possible to equate that part of the wavefront to awavefront represented within the constricted pupil to obtain arepresentation of a generic formula. Hence, from the definition ofpolynomials it is possible to obtain a pupil scaling factor epsilon andpupil radius rho. For polynomials that can be separated into radialpolynomials and angular component, it is possible ignore the angularcomponent and assume the constriction is concentric. Hence, it ispossible to determine a generic pupil rescaling formula (GPRF) for anybasis function that can be separated into radial polynomials and atriangular function. In some cases, the GPRF can be defined as theproduct of a pupil rescaling/resizing polynomial factor, and a radialpolynomial, where the radial polynomial is determined prior to resizing.Embodiments of the present invention provide pupil resizing polynomialsfor Zernike basis functions, Taylor basis functions, Seidel basisfunctions, and the like, and methods and system for obtaining such pupilresizing polynomials and for using the same for resizing purposes.Embodiments also encompass methods and systems for calculating ordetermining refractions based on a new set of polynomials, after ageometrical transformation such as a pupil constriction, a rotations, ora decentration.

The present invention can be readily adapted for use with existing lasersystems, wavefront measurement systems, and other optical measurementdevices. Although the systems, software, and methods of the presentinvention are described primarily in the context of a laser eye surgerysystem, it should be understood the present invention may be adapted foruse in alternative eye treatment procedures, systems, or modalities,such as spectacle lenses, intraocular lenses, accommodating IOLs,contact lenses, corneal ring implants, collagenous corneal tissuethermal remodeling, corneal inlays, corneal onlays, other cornealimplants or grafts, and the like. Relatedly, systems, software, andmethods according to embodiments of the present invention are wellsuited for customizing any of these treatment modalities to a specificpatient. Thus, for example, embodiments encompass custom intraocularlenses, custom contact lenses, custom corneal implants, and the like,which can be configured to treat or ameliorate any of a variety ofvision conditions in a particular patient based on their unique ocularcharacteristics or anatomy.

Turning now to the drawings, FIG. 1 illustrates a laser eye surgerysystem 10 of the present invention, including a laser 12 that produces alaser beam 14. Laser 12 is optically coupled to laser delivery optics16, which directs laser beam 14 to an eye E of patient P. A deliveryoptics support structure (not shown here for clarity) extends from aframe 18 supporting laser 12. A microscope 20 is mounted on the deliveryoptics support structure, the microscope often being used to image acornea of eye E.

Laser 12 generally comprises an excimer laser, ideally comprising anargon-fluorine laser producing pulses of laser light having a wavelengthof approximately 193 nm. Laser 12 will preferably be designed to providea feedback stabilized fluence at the patient's eye, delivered viadelivery optics 16. The present invention may also be useful withalternative sources of ultraviolet or infrared radiation, particularlythose adapted to controllably ablate the corneal tissue without causingsignificant damage to adjacent and/or underlying tissues of the eye.Such sources include, but are not limited to, solid state lasers andother devices which can generate energy in the ultraviolet wavelengthbetween about 185 and 205 nm and/or those which utilizefrequency-multiplying techniques. Hence, although an excimer laser isthe illustrative source of an ablating beam, other lasers may be used inthe present invention.

Laser system 10 will generally include a computer or programmableprocessor 22. Processor 22 may comprise (or interface with) aconventional PC system including the standard user interface devicessuch as a keyboard, a display monitor, and the like. Processor 22 willtypically include an input device such as a magnetic or optical diskdrive, an internet connection, or the like. Such input devices willoften be used to download a computer executable code from a tangiblestorage media 29 embodying any of the methods of the present invention.Tangible storage media 29 may take the form of a floppy disk, an opticaldisk, a data tape, a volatile or non-volatile memory, RAM, or the like,and the processor 22 will include the memory boards and other standardcomponents of modern computer systems for storing and executing thiscode. Tangible storage media 29 may optionally embody wavefront sensordata, wavefront gradients, a wavefront elevation map, a treatment map, acorneal elevation map, and/or an ablation table. While tangible storagemedia 29 will often be used directly in cooperation with a input deviceof processor 22, the storage media may also be remotely operativelycoupled with processor by means of network connections such as theinternet, and by wireless methods such as infrared, Bluetooth, or thelike.

Laser 12 and delivery optics 16 will generally direct laser beam 14 tothe eye of patient P under the direction of a computer 22. Computer 22will often selectively adjust laser beam 14 to expose portions of thecornea to the pulses of laser energy so as to effect a predeterminedsculpting of the cornea and alter the refractive characteristics of theeye. In many embodiments, both laser beam 14 and the laser deliveryoptical system 16 will be under computer control of processor 22 toeffect the desired laser sculpting process, with the processor effecting(and optionally modifying) the pattern of laser pulses. The pattern ofpulses may by summarized in machine readable data of tangible storagemedia 29 in the form of a treatment table, and the treatment table maybe adjusted according to feedback input into processor 22 from anautomated image analysis system in response to feedback data providedfrom an ablation monitoring system feedback system. Optionally, thefeedback may be manually entered into the processor by a systemoperator. Such feedback might be provided by integrating the wavefrontmeasurement system described below with the laser treatment system 10,and processor 22 may continue and/or terminate a sculpting treatment inresponse to the feedback, and may optionally also modify the plannedsculpting based at least in part on the feedback. Measurement systemsare further described in U.S. Pat. No. 6,315,413, the full disclosure ofwhich is incorporated herein by reference.

Laser beam 14 may be adjusted to produce the desired sculpting using avariety of alternative mechanisms. The laser beam 14 may be selectivelylimited using one or more variable apertures. An exemplary variableaperture system having a variable iris and a variable width slit isdescribed in U.S. Pat. No. 5,713,892, the full disclosure of which isincorporated herein by reference. The laser beam may also be tailored byvarying the size and offset of the laser spot from an axis of the eye,as described in U.S. Pat. Nos. 5,683,379, 6,203,539, and 6,331,177, thefull disclosures of which are incorporated herein by reference.

Still further alternatives are possible, including scanning of the laserbeam over the surface of the eye and controlling the number of pulsesand/or dwell time at each location, as described, for example, by U.S.Pat. No. 4,665,913, the full disclosure of which is incorporated hereinby reference; using masks in the optical path of laser beam 14 whichablate to vary the profile of the beam incident on the cornea, asdescribed in U.S. Pat. No. 5,807,379, the full disclosure of which isincorporated herein by reference; hybrid profile-scanning systems inwhich a variable size beam (typically controlled by a variable widthslit and/or variable diameter iris diaphragm) is scanned across thecornea; or the like. The computer programs and control methodology forthese laser pattern tailoring techniques are well described in thepatent literature.

Additional components and subsystems may be included with laser system10, as should be understood by those of skill in the art. For example,spatial and/or temporal integrators may be included to control thedistribution of energy within the laser beam, as described in U.S. Pat.No. 5,646,791, the full disclosure of which is incorporated herein byreference. Ablation effluent evacuators/filters, aspirators, and otherancillary components of the laser surgery system are known in the art.Further details of suitable systems for performing a laser ablationprocedure can be found in commonly assigned U.S. Pat. Nos. 4,665,913,4,669,466, 4,732,148, 4,770,172, 4,773,414, 5,207,668, 5,108,388,5,219,343, 5,646,791 and 5,163,934, the complete disclosures of whichare incorporated herein by reference. Suitable systems also includecommercially available refractive laser systems such as thosemanufactured and/or sold by Alcon, Bausch & Lomb, Nidek, WaveLight,LaserSight, Schwind, Zeiss-Meditec, and the like. Basis data can befurther characterized for particular lasers or operating conditions, bytaking into account localized environmental variables such astemperature, humidity, airflow, and aspiration.

FIG. 2 is a simplified block diagram of an exemplary computer system 22that may be used by the laser surgical system 10 of the presentinvention. Computer system 22 typically includes at least one processor52 which may communicate with a number of peripheral devices via a bussubsystem 54. These peripheral devices may include a storage subsystem56, comprising a memory subsystem 58 and a file storage subsystem 60,user interface input devices 62, user interface output devices 64, and anetwork interface subsystem 66. Network interface subsystem 66 providesan interface to outside networks 68 and/or other devices, such as thewavefront measurement system 30.

User interface input devices 62 may include a keyboard, pointing devicessuch as a mouse, trackball, touch pad, or graphics tablet, a scanner,foot pedals, a joystick, a touchscreen incorporated into the display,audio input devices such as voice recognition systems, microphones, andother types of input devices. User input devices 62 will often be usedto download a computer executable code from a tangible storage media 29embodying any of the methods of the present invention. In general, useof the term “input device” is intended to include a variety ofconventional and proprietary devices and ways to input information intocomputer system 22.

User interface output devices 64 may include a display subsystem, aprinter, a fax machine, or non-visual displays such as audio outputdevices. The display subsystem may be a cathode ray tube (CRT), aflat-panel device such as a liquid crystal display (LCD), a projectiondevice, or the like. The display subsystem may also provide a non-visualdisplay such as via audio output devices. In general, use of the term“output device” is intended to include a variety of conventional andproprietary devices and ways to output information from computer system22 to a user.

Storage subsystem 56 can store the basic programming and data constructsthat provide the functionality of the various embodiments of the presentinvention. For example, a database and modules implementing thefunctionality of the methods of the present invention, as describedherein, may be stored in storage subsystem 56. These software modulesare generally executed by processor 52. In a distributed environment,the software modules may be stored on a plurality of computer systemsand executed by processors of the plurality of computer systems. Storagesubsystem 56 typically comprises memory subsystem 58 and file storagesubsystem 60.

Memory subsystem 58 typically includes a number of memories including amain random access memory (RAM) 70 for storage of instructions and dataduring program execution and a read only memory (ROM) 72 in which fixedinstructions are stored. File storage subsystem 60 provides persistent(non-volatile) storage for program and data files, and may includetangible storage media 29 (FIG. 1) which may optionally embody wavefrontsensor data, wavefront gradients, a wavefront elevation map, a treatmentmap, and/or an ablation table. File storage subsystem 60 may include ahard disk drive, a floppy disk drive along with associated removablemedia, a Compact Digital Read Only Memory (CD-ROM) drive, an opticaldrive, DVD, CD-R, CD-RW, solid-state removable memory, and/or otherremovable media cartridges or disks. One or more of the drives may belocated at remote locations on other connected computers at other sitescoupled to computer system 22. The modules implementing thefunctionality of the present invention may be stored by file storagesubsystem 60.

Bus subsystem 54 provides a mechanism for letting the various componentsand subsystems of computer system 22 communicate with each other asintended. The various subsystems and components of computer system 22need not be at the same physical location but may be distributed atvarious locations within a distributed network. Although bus subsystem54 is shown schematically as a single bus, alternate embodiments of thebus subsystem may utilize multiple busses.

Computer system 22 itself can be of varying types including a personalcomputer, a portable computer, a workstation, a computer terminal, anetwork computer, a control system in a wavefront measurement system orlaser surgical system, a mainframe, or any other data processing system.Due to the ever-changing nature of computers and networks, thedescription of computer system 22 depicted in FIG. 2 is intended only asa specific example for purposes of illustrating one embodiment of thepresent invention. Many other configurations of computer system 22 arepossible having more or less components than the computer systemdepicted in FIG. 2.

Referring now to FIG. 3, one embodiment of a wavefront measurementsystem 30 is schematically illustrated in simplified form. In verygeneral terms, wavefront measurement system 30 is configured to senselocal slopes of a gradient map exiting the patient's eye. Devices basedon the Hartmann-Shack principle generally include a lenslet array tosample the gradient map uniformly over an aperture, which is typicallythe exit pupil of the eye. Thereafter, the local slopes of the gradientmap are analyzed so as to reconstruct the wavefront surface or map.

More specifically, one wavefront measurement system 30 includes an imagesource 32, such as a laser, which projects a source image throughoptical tissues 34 of eye E so as to form an image 44 upon a surface ofretina R. The image from retina R is transmitted by the optical systemof the eye (e.g., optical tissues 34) and imaged onto a wavefront sensor36 by system optics 37. The wavefront sensor 36 communicates signals toa computer system 22′ for measurement of the optical errors in theoptical tissues 34 and/or determination of an optical tissue ablationtreatment program. Computer 22′ may include the same or similar hardwareas the computer system 22 illustrated in FIGS. 1 and 2. Computer system22′ may be in communication with computer system 22 that directs thelaser surgery system 10, or some or all of the components of computersystem 22, 22′ of the wavefront measurement system 30 and laser surgerysystem 10 may be combined or separate. If desired, data from wavefrontsensor 36 may be transmitted to a laser computer system 22 via tangiblemedia 29, via an I/O port, via an networking connection 66 such as anintranet or the Internet, or the like.

Wavefront sensor 36 generally comprises a lenslet array 38 and an imagesensor 40. As the image from retina R is transmitted through opticaltissues 34 and imaged onto a surface of image sensor 40 and an image ofthe eye pupil P is similarly imaged onto a surface of lenslet array 38,the lenslet array separates the transmitted image into an array ofbeamlets 42, and (in combination with other optical components of thesystem) images the separated beamlets on the surface of sensor 40.Sensor 40 typically comprises a charged couple device or “CCD,” andsenses the characteristics of these individual beamlets, which can beused to determine the characteristics of an associated region of opticaltissues 34. In particular, where image 44 comprises a point or smallspot of light, a location of the transmitted spot as imaged by a beamletcan directly indicate a local gradient of the associated region ofoptical tissue.

Eye E generally defines an anterior orientation ANT and a posteriororientation POS. Image source 32 generally projects an image in aposterior orientation through optical tissues 34 onto retina R asindicated in FIG. 3. Optical tissues 34 again transmit image 44 from theretina anteriorly toward wavefront sensor 36. Image 44 actually formedon retina R may be distorted by any imperfections in the eye's opticalsystem when the image source is originally transmitted by opticaltissues 34. Optionally, image source projection optics 46 may beconfigured or adapted to decrease any distortion of image 44.

In some embodiments, image source optics 46 may decrease lower orderoptical errors by compensating for spherical and/or cylindrical errorsof optical tissues 34. Higher order optical errors of the opticaltissues may also be compensated through the use of an adaptive opticelement, such as a deformable mirror (described below). Use of an imagesource 32 selected to define a point or small spot at image 44 uponretina R may facilitate the analysis of the data provided by wavefrontsensor 36. Distortion of image 44 may be limited by transmitting asource image through a central region 48 of optical tissues 34 which issmaller than a pupil 50, as the central portion of the pupil may be lessprone to optical errors than the peripheral portion. Regardless of theparticular image source structure, it will be generally be beneficial tohave a well-defined and accurately formed image 44 on retina R.

In one embodiment, the wavefront data may be stored in a computerreadable medium 29 or a memory of the wavefront sensor system 30 in twoseparate arrays containing the x and y wavefront gradient valuesobtained from image spot analysis of the Hartmann-Shack sensor images,plus the x and y pupil center offsets from the nominal center of theHartmann-Shack lenslet array, as measured by the pupil camera 51 (FIG.3) image. Such information contains all the available information on thewavefront error of the eye and is sufficient to reconstruct thewavefront or any portion of it. In such embodiments, there is no need toreprocess the Hartmann-Shack image more than once, and the data spacerequired to store the gradient array is not large. For example, toaccommodate an image of a pupil with an 8 mm diameter, an array of a20×20 size (i.e., 400 elements) is often sufficient. As can beappreciated, in other embodiments, the wavefront data may be stored in amemory of the wavefront sensor system in a single array or multiplearrays.

While the methods of the present invention will generally be describedwith reference to sensing of an image 44, a series of wavefront sensordata readings may be taken. For example, a time series of wavefront datareadings may help to provide a more accurate overall determination ofthe ocular tissue aberrations. As the ocular tissues can vary in shapeover a brief period of time, a plurality of temporally separatedwavefront sensor measurements can avoid relying on a single snapshot ofthe optical characteristics as the basis for a refractive correctingprocedure. Still further alternatives are also available, includingtaking wavefront sensor data of the eye with the eye in differingconfigurations, positions, and/or orientations. For example, a patientwill often help maintain alignment of the eye with wavefront measurementsystem 30 by focusing on a fixation target, as described in U.S. Pat.No. 6,004,313, the full disclosure of which is incorporated herein byreference. By varying a position of the fixation target as described inthat reference, optical characteristics of the eye may be determinedwhile the eye accommodates or adapts to image a field of view at avarying distance and/or angles.

The location of the optical axis of the eye may be verified by referenceto the data provided from a pupil camera 52. In the exemplaryembodiment, a pupil camera 52 images pupil 50 so as to determine aposition of the pupil for registration of the wavefront sensor datarelative to the optical tissues.

An alternative embodiment of a wavefront measurement system isillustrated in FIG. 3A. The major components of the system of FIG. 3Aare similar to those of FIG. 3. Additionally, FIG. 3A includes anadaptive optical element 53 in the form of a deformable mirror. Thesource image is reflected from deformable mirror 98 during transmissionto retina R, and the deformable mirror is also along the optical pathused to form the transmitted image between retina R and imaging sensor40. Deformable mirror 98 can be controllably deformed by computer system22 to limit distortion of the image formed on the retina or ofsubsequent images formed of the images formed on the retina, and mayenhance the accuracy of the resultant wavefront data. The structure anduse of the system of FIG. 3A are more fully described in U.S. Pat. No.6,095,651, the full disclosure of which is incorporated herein byreference.

The components of an embodiment of a wavefront measurement system formeasuring the eye and ablations may comprise elements of a WaveScan®system, available from VISX, INCORPORATED of Santa Clara, Calif. Oneembodiment includes a WaveScan system with a deformable mirror asdescribed above. An alternate embodiment of a wavefront measuring systemis described in U.S. Pat. No. 6,271,915, the full disclosure of which isincorporated herein by reference. It is appreciated that any wavefrontaberrometer could be employed for use with the present invention.Relatedly, embodiments of the present invention encompass theimplementation of any of a variety of optical instruments provided byWaveFront Sciences, Inc., including the COAS wavefront aberrometer, theClearWave contact lens aberrometer, the CrystalWave IOL aberrometer, andthe like. Embodiments of the present invention may also involvewavefront measurement schemes such as a Tscherning-based system, whichmay be provided by WaveFront Sciences, Inc. Embodiments of the presentinvention may also involve wavefront measurement schemes such as a raytracing-based system, which may be provided by Tracey Technologies,Corp.

Ocular wavefront transformation is suitable for use in wavefront opticsfor vision correction because the pupil size of a human eye oftenchanges due to accommodation or the change of lighting, and because thepupil constriction is commonly not concentric. Certain features of theseocular effects are discussed in, for example, Wilson, M. A. et al.,Optom. Vis. Sci., 69:129-136 (1992), Yang, Y. et al., Invest. Ophthal.Vis. Sci., 43:2508-2512 (2002), and Donnenfeld, E. J., Refract. Surg.,20:593-596 (2004). For example, in laser vision correction, the pupilsize of an eye is relatively large when an ocular wavefront is capturedunder an aberrometer. To obtain the entire ocular wavefront, it is oftenrecommended that the ambient light be kept low so as to dilate the pupilsize during the wavefront exam. A larger wavefront map can providesurgeons the flexibility for treatment over a smaller zone, because thewavefront information over any smaller zone within a larger zone isknown. However, if a smaller wavefront map is captured, it may bedifficult or impossible to devise an accurate treatment over a largerzone, because the wavefront information outside of the captured zone isunknown. When the patient is under the laser, the pupil size can changedue to changes in the ambient light. In many cases, the surgery room isbrighter than a wavefront examination room, in particular when thepatient is under the hood. Furthermore, the cyclorotation of the eye dueto the change from a sitting position to a laying position can make thepupil center change between the wavefront capture and the laserablation, for example as discussed in Chernyak, D. A., J. Cataract.Refract. Surg., 30:633-638 (2004). Theoretically, it has been reportedthat correction of error due to rotation and translation of the pupilcan provide significant benefits in vision correction. Certain aspectsof these ocular effects are discussed in Bará, S. et al., Appl. Opt.,39:3413-3420 (2000) and Guirao, A. et al., J. Opt. Soc. Am. A,18:1003-1015 (2001).

Iris registration, as discussed for example in Chernyak, D. A., J.Refract. Surg., 21:463-468 (2005), can be used to correct or reduce theerror from the misalignment between the pupil in front of theaberrometer and the pupil under the laser. Because the iris features aretypically not affected by the change of pupil size, they can be used asreliable references to establish the relative geometrical displacementbetween two image frames, as discussed in Daugman, J., IEEE Trans, PAMI,15:1148-1161 (1993). A common coordinate system can thus be establishedso as to facilitate the full correction of ocular aberrations. Forpractical applications, however, a full correction may not be possiblepartly because of the fluctuation of the high order aberrations andpartly because of the instrument error. Therefore, it may be useful tohave a tool for the error analysis of an imperfect correction for themisalignment of the eye between the pupil in front of the aberrometerand the pupil under the laser. Embodiments of the present inventionprovide systems and methods for predicting error if no registration isperformed, or if registration is inaccurately performed. Moreover, for amajority of the data analysis for ocular aberrations, it is oftenhelpful to standardize pupil sizes of different wavefront exams to agiven pupil size. Embodiments of the present invention encompass pupilresizing of known wavefronts. In addition, the constriction anddecentration of a pupil can lead to wavefront refraction change whenhigh order aberrations are present. Certain aspects of this oculareffect can be used as the basis for designing optical surfaces for thecorrection or treatment of presbyopia, a condition which is discussed inDai, G-m., Appl. Opt., 45:4184-4195 (2006).

1. Wavefront Transformation and Iris Registration

In understanding wavefront transformation and iris registration, it ishelpful to consider features of a human eye and how an iris registrationis implemented.

1.1 Definitions

The following exemplary definitions may be useful for a discussion ofwavefront transformation and iris registration for vision correction,according to some embodiments of the present invention. FIG. 4 shows anillustration of the human eye 400, and depicts the following features:optical axis, visual axis, pupillary axis, angle alpha, angle kappa(angle lambda), and corneal vertex (not to scale). N and N′ are thefirst and second nodal points, and E and E′ are the centers of theentrance and exit pupils, respectively.

Purkinje images can be defined as images of a light source reflected bydifferent surfaces of the optics of the eye. A first Purkinje image(Purkinje I) can be the reflection from the anterior surface of thecornea. A second Purkinje image (Purkinje II) can be the reflection fromthe posterior surface of the cornea. A third Purkinje image (PurkinjeIII) can be the reflection of the anterior surface of the crystallinelens. A fourth Purkinje image (Purkinje IV) can be the reflection of theposterior surface of the crystalline lens and can be the only invertedimage. The brightness of the Purkinje images can be calculated from theFresnel equation.

The optical axis 410 of a human eye can be defined as an imaginary linethat connects a point source and all Purkinje images when they arealigned to coincide. Because the eye is typically not rotationallysymmetric, this alignment of all Purkinje images may be difficult toachieve.

The visual axis 420 of a human eye can be defined as a line thatconnects the light source and first nodal point (N) and the second nodalpoint (N′) to the fovea when the eye is fixated to the target. Thevisual axis can also be referred to as the line of sight.

The pupillary axis 430 of a human eye can be defined as the line that isperpendicular to the cornea and connects to the center of the entrancepupil (E) and the center of the exit pupil (E′) to the fovea. In someembodiments, this can be achieved by adjusting the first Purkinje imageto the center of the entrance pupil so the line connecting the lightsource and the pupil center defines the pupillary axis.

Angle Kappa 440 can be defined as the angle between the pupillary axisand visual axis, or the line of sight. Angle kappa may also be referredto as angle lambda. Angle kappa can be defined as positive if thepupillary axis is nasal to the visual axis, and negative if it istemporal to the visual axis. Typically, the angle kappa is smaller thanthe angle alpha.

Angle Alpha 450 can be defined as the angle between the visual axis andthe optical axis. A typical value of angle alpha can be within a rangefrom about 4°≦α≦8°.

The corneal apex 460 can be defined as the point on the cornea that hasthe steepest curvature. For example, the corneal apex 460 can bedisposed at the intersection of the anterior surface of the cornea andthe optical axis. In some embodiments, it is a fixed point to a givencornea and does not depend upon any measurements. The corneal apex cansometimes be confused with the corneal vertex.

The corneal vertex 470 can be defined as the intersection of thepupillary axis with the anterior surface of the cornea, if the pupillaryaxis coincides with the optical axis of the measuring device, such as acorneal topographer.

The pupil center 480 can be defined as the center of a best fit ellipseto the pupil. The majority of human pupils are elliptical to someextent. Some pupils are even irregular.

As an exemplary illustration, it is possible to estimate the distance onthe cornea for a kappa angle of 3.5° as follows. Using a nominal valueof 3.5 mm as the anterior chamber depth, we obtain3.5×tan(3.5π/180)=0.214 mm. Therefore, in this example the cornealvertex is two tenths of a millimeter nasal to the pupil center.

1.2 Iris Registration

In understanding iris registration, it is helpful to consider a typicalsituation for wavefront-driven refractive surgery as shown in FIGS. 5Aand 5B. The patient is brought in for pre-operatively wavefront exam infront of a wavefront aberrometer. In some embodiments, to capture theentire ocular aberration of the eye, the wavefront measurement room isusually dimmed to scotopic conditions. As such, the pupil size isrelatively large. In some embodiments, when the patient is laying underthe laser, the surgery room is relatively bright so the pupil constrictsto a smaller size. In general, the pupil constriction is not concentric.Therefore, the pupil center can shift between these two situations withrespect to a stationary reference, such as the iris of the eye. FIG. 5Aprovides an exemplary illustration of a human eye when the patient is infront of the wavefront device. This may correspond to an evaluationenvironment or context. FIG. 5B provides an exemplary illustration of ahuman eye when the patient is under the laser (not to scale). This maycorrespond to a treatment environment or context. As shown here, an eye500 can present an iris center 510, a pupil center 520 when the patientis in front of the wavefront device, a pupil center 530 when the patientis under the laser, an iris boundary 540, and one or more iris features550. A distance between the two pupil centers 520, 530 can be referredto as a pupil center shift 560.

When the ocular wavefront is examined, a treatment plan is typicallygenerated based on the ocular aberrations. If a treatment is referencedto the pupil center, it may not be delivered to the correct location ifthe pupil center shifts, as can be seen in FIGS. 5A and 5B. The iris ofthe human eye contains irregular texture that can be used as coordinatereferences, because the iris (together with the texture) typically doesnot change when the pupil size changes. Hence, in an exemplary approacha certain number of iris features can be identified and used asreferences. A treatment plan can be referenced to the stable irisfeatures when the plan is created. When the patient is laying under thelaser, the eye of the patient can be captured and analyzed. The irisfeatures can be identified again and the coordinate can be established.The laser delivery optics are aligned properly so the two coordinatesystems coincide. Consequently, the treatment can be delivered correctlyas planned.

One of the side results may be a determination of the corneal vertexfrom the first Purkinje image of the laser source of the wavefrontdevice, as shown in FIG. 6. As seen in this exemplary diagram of an eye600, a pupil center 610 and a corneal vertex 620 are the twointersections of the visual axis and the pupillary axis, respectively,with the anterior surface of the cornea. Therefore, the distance betweenthe pupil center and the corneal vertex can determine the angle kappa onthe anterior surface of the cornea. Although the visual axis may notstrictly pass through the pupil center, the deviation can be very smalland often negligible. FIG. 6 presents a pupil image that shows an iriscenter 630, pupil center 610, and corneal vertex 620 that is thePurkinje reflex of the laser source of the wavefront device. Both theiris boundary and the pupil boundary can be detected with best-fitellipses.

For the correction of presbyopia, which is discussed for example in Dai,G-m., Appl. Opt., 45:4184-4195 (2006), some surgeons believe that it isbetter to put the presbyopic correction shape over the corneal vertexinstead of the pupil center as the pupil center can tend to move towardthe corneal vertex during accommodation. Some studies, including Yang,Y. et al., Invest. Ophthal. Vis. Sci., 43:2508-2512 (2002), Walsh, G.,Ophthal. Physiol. Opt., 8:178-182 (1988), and Wyatt, H. J., Vis. Res.,35:2021-2036 (1995) have indicated that the pupil center tends to movenasally and inferiorly when the pupil constricts. It has now beendiscovered that there is a weak but statistically significantcorrelation between the pupil center shift and the angle kappa in the xdirection. Embodiments of the present invention encompass systems andmethods for putting a presbyopic correction shape over an accommodatedpupil center, rather than putting it over the corneal vertex.

2. Wavefront Representation for Pupil Resizing

As discussed elsewhere herein, a pupil can constrict because of anincrease of the ambient lighting and because of accommodation. Forwavefront analysis, a commonly used metric involves the root mean square(RMS) error of the wavefront. However, the RMS wavefront error typicallydepends upon the pupil size, or more strictly speaking, the wavefrontdiameter. Therefore, it can be helpful to normalize (or resize)wavefront maps to the same pupil size.

Typically, pupil constriction is not concentric. According to someembodiments of the present invention, the pupil constriction can betreated as concentric. A discussion of non-concentric pupil constrictionis further discussed in Section 5, below. For the majority of wavefrontanalysis, the pupil resizing does not involve a large amount of pupilsize changes. For example, for non-presbyopic eyes, a 6 mm pupil size isoften used as a normalized pupil size; for presbyopic eyes, a 5 mm pupilsize can be used instead. The pupil center shift due to the pupil sizechange under these conditions is relatively small and may be ignored formost of the analysis. In the case where a more accurate analysis isneeded or desired, it is helpful to refer to the discussion in section5, below.

2.1 General Consideration

A discussion of wavefront representation for pupil constriction isprovided in Dai, G.-m., J. Opt. Soc. Am. A., 23:539-543 (2006), whenZernike polynomials are used as the basis functions. It may be assumedthat optical properties of human eye do not change when pupil sizechanges. A resizing technique has now been discovered that can be usedwith any basis functions. Suppose an ocular wavefront is represented bya set of basis functions {F_(i)(ρ,θ)} as

$\begin{matrix}{{W\left( {{R_{1}\rho},\theta} \right)} = {\sum\limits_{i = 0}^{J}{a_{i}{{F_{i}\left( {\rho,\theta} \right)}.}}}} & (1)\end{matrix}$where R₁ is the pupil radius, J is the highest basis function, and a_(i)is the coefficient of the ith basis function. We further assume that{F_(i)(ρ,θ)} can be separated into a set of radial polynomials and atriangular function asF _(i)(ρ,θ)=S _(i)(ρ)T _(i)(θ).   (2)

FIG. 7A provides a contour plots of a wavefront map 700 a with pupilradius R1 and FIG. 7B provides a contour plot of the wavefront map 700 bwhen the pupil size constricts to pupil radius R2. Both maps are in thesame scale, and units can be in microns of optical path difference. Theportion of the wavefront defined by R2 in FIG. 7A is the same as theplot in FIG. 7B. Consider for example an ocular wavefront of 6 mm pupil,which may be illustrated by FIG. 7A. When the pupil constricts to R₂,only the part that is within radius R₂ is represented, as may beillustrated in FIG. 7B. Because the optical components, which are oftenmainly the cornea and the crystalline lens, typically do not changeduring the pupil constriction, the aberration pattern of the constrictedwavefront shown in FIG. 7B is the same as the original wavefront withinradius R₂ as shown in FIG. 7A. When ρ=1, W(R₁ρ,θ) represents the entirewavefront. When ρ becomes smaller than 1, the represented wavefrontbecomes smaller. Hence, the part of the wavefront within radius R₂ inFIG. 7A can be expressed as W(R₂ρ,θ), or expressed as W(R₁ερ,θ) bysimply scaling the radial variable ρ by ε=R₂/R₁ to ερ. Therefore, wehaveW(R ₁ερ,θ)=W(R ₂ρ,θ).   (3)For the wavefront as shown in FIG. 7B, we can represent it as

$\begin{matrix}{{{W\left( {{R_{2}\rho},\theta} \right)} = {\sum\limits_{i = 0}^{J}{b_{i}{F_{i}\left( {\rho,\theta} \right)}}}},} & (4)\end{matrix}$where b_(i) is the coefficient of the ith basis function. SubstitutingEqs. (1) and (6) into (3), we get

$\begin{matrix}{{\sum\limits_{i = 0}^{J}{a_{i}{F_{i}\left( {{ɛ\rho},\theta} \right)}}} = {\sum\limits_{i = 0}^{J}{b_{i}{{F_{i}\left( {\rho,\theta} \right)}.}}}} & (5)\end{matrix}$

Substituting Eq. (2) into Eq. (5) and considering the fact that thetriangular function T_(i)(θ) can be the same on both sides of Eq. (5)because no rotation is involved, we obtain

$\begin{matrix}{{\sum\limits_{i = 0}^{J}{a_{i}{S_{i}({ɛ\rho})}}} = {\sum\limits_{i = 0}^{J}{b_{i}{{S_{i}(\rho)}.}}}} & (6)\end{matrix}$

Equation (6) is the basis for relating the coefficients of a set ofbasis functions before and after pupil constriction. It can apply to anyset of basis functions so long as the basis set can be separated into aproduct of a set of radial polynomials and a triangular function.

2.2 Pupil Resizing Polynomials

Suppose the radial polynomials S_(i)(ρ) is orthogonal over the unitcircle and the orthogonality is written as

$\begin{matrix}{{\frac{1}{A}{\int_{0}^{1}{{S_{i}(\rho)}{S_{i^{\prime}}(\rho)}\rho{\mathbb{d}\rho}}}} = {\delta_{{ii}^{\prime}}.}} & (7)\end{matrix}$

In Eq. (7), A is an orthogonalization constant. Multiplying S_(i′)(ρ) onboth sides of Eq. (6), integrating over the unit circle, and using theorthogonality in Eq. (7), we have

$\begin{matrix}\begin{matrix}{b_{i^{\prime}} = {\sum\limits_{i = 0}^{J}\;{a_{i}{\int_{0}^{1}{{S_{i}\left( {ɛ\;\rho} \right)}{S_{i^{\prime}}(\rho)}\rho\ {\mathbb{d}\rho}}}}}} \\{{= {\sum\limits_{i = 0}^{J}\;{{\mathcal{H}_{i^{\prime}i}(ɛ)}a_{i}}}},}\end{matrix} & (8)\end{matrix}$where the pupil resizing polynomials H_(i′i)(ε) can be expressed as

_(i′i)(ε)=∫₀ ¹ S _(i)(ερ)S _(i′)(ρ)ρdρ.   (9)

Aspects of equation (9) are discussed in Janssen, A. J. E. M., J.Microlith., Microfab., Microsyst., 5:030501 (2006). It has now beendiscovered that equation (9) can be applied to any set of basisfunctions of which the radial polynomials are orthogonal.

When the set of radial polynomials {S_(i)(ρ)} is not orthogonal, adifferent approach can be used. Because the radial polynomials{S_(i)(ρ)} are polynomials of ρ, we may write S_(i)(ρ) as

$\begin{matrix}{{{S_{i}(\rho)} = {\sum\limits_{k = 0}^{i}\;{h_{k}\rho^{k}}}},} & (10)\end{matrix}$where h_(k) is the kth polynomial coefficient that depends only upon theindex k. Equation (10) indicates that the variables ε and ρ areseparable in the set of radial polynomials S_(i)(ερ) as

$\begin{matrix}{{S_{i}({ɛ\rho})} = {\sum\limits_{k = 0}^{i}\;{{\mathcal{H}_{ki}(ɛ)}{{S_{i}(\rho)}.}}}} & (11)\end{matrix}$Substituting Eqs. (10) and (11) into Eq. (6), we have

$\begin{matrix}{{\sum\limits_{i = 0}^{J}\;{a_{i}{\sum\limits_{k = 0}^{i}\;{{H_{ki}(ɛ)}{S_{i}(\rho)}}}}} = {\sum\limits_{i = 0}^{J}\;{b_{i}{{S_{i}(\rho)}.}}}} & (12)\end{matrix}$Since S_(i)(ρ) appears on both sides of Eq. (12), it can be eliminatedso that Eq. (12) is simplified as

$\begin{matrix}{{b_{i}(\rho)} = {\sum\limits_{k = 0}^{i}\;{{\mathcal{H}_{ki}(ɛ)}{a_{i}.}}}} & (13)\end{matrix}$

Equation (13) gives a general expression of a new set of coefficients asrelated to an original set of coefficients when the pupil size changes.The set of polynomials

_(k)(ε) is termed the pupil resizing polynomials that is useful in thecalculation of coefficients of basis functions when the pupil isresized. Equation (6) presents a generic formula, or a basis for pupilrescaling. Equations (9) and (13) present two different methods of theapproach.

Hence, embodiments of the present invention encompass pupil resizingpolynomials for several useful sets of basis functions, including Taylormonomials, Zernike polynomials, and Seidel series.

2.3 Taylor Resizing Monomials

When a wavefront is represented by Taylor monomials, the set of Taylorcoefficients changes accordingly when the pupil size changes. Taylormonomials can be written as a product of the radial power and thetriangular function asT _(p) ^(q)(ρ,θ)=ρ^(p) cos^(q) θ sin^(p-q) θ.   (14)Therefore, the radial monomials can be written asS _(p)(ρ)=ρ^(p).   (15)Substituting Eq. (15) into Eq. (11), we haveS _(ρ)(ερ)=ε^(ρ)ρ^(ρ)=ε^(ρ) S _(ρ)(ρ).   (16)Hence, the Taylor resizing monomials can be expressed asL _(ρ)(ε)=ε^(ρ).   (17)

Equation (17) indicates that the set of Taylor resizing monomials is aset of power series of the pupil resizing ratio ε. In other words, eachnew Taylor coefficient is scaled by ε^(ρ) where ρ is the radial degreeof the Taylor monomial. Equation (17) can be a GPRF for a Taylor basisfunction. The triangular function discussed here is similar to thetriangular function discussed for the Zernike polynomials.

As an example, Table 1 shows a set of Taylor coefficients and thecorresponding resized Taylor coefficients when a pupil resizing ratio of0.8 is assumed. The original wavefront map 800 a shown in FIG. 8A andthe resized wavefront map 800 b shown in FIG. 8B correspond tocoefficients listed in Table 1. The resized wavefront appears identicalto the inner part of the original wavefront within the new pupil size.

Table 1 shows Taylor coefficients before (a_(ρ) ^(q)) and after (b_(ρ)^(q)) pupil constriction, where ε=0.8.

TABLE 1 i p q a_(ρ) ^(q) b_(ρ) ^(q) 0 0 0 1.0660 1.0660 1 1 0 2.63352.1068 2 1 1 −3.1810 2.1068 3 2 0 −4.6450 −2.9728 4 2 1 4.0090 2.5658 52 2 −4.3256 −2.2147 6 3 0 −1.6533 −0.8465 7 3 1 16.4753 8.4354 8 3 21.4026 0.7181 9 3 3 6.9912 2.8636 10 4 0 −1.2680 −0.5194 11 4 1 4.79391.9636 12 4 2 13.3486 5.4676 13 4 3 −0.5777 −0.2366 14 4 4 8.5984 2.817515 5 0 1.2909 0.4230 16 5 1 −15.7024 −5.1454 17 5 5 −6.0772 −1.9914 18 53 −19.7837 −6.4827 19 5 4 −3.7889 −1.2415 20 5 5 −2.5517 −0.6689 21 6 04.2625 1.1174 22 6 1 −7.2498 −1.9005 23 6 2 2.7658 0.7250 24 6 3−10.5176 −2.7571 25 6 4 −15.8385 −4.1520 26 6 5 −6.3212 −1.6571 27 6 6−5.4349 −1.1398

2.4 Zernike Resizing Polynomials

Techniques for calculating a new set of Zernike coefficients from anoriginal set when the pupil size changes has been investigated by anumber of authors. For example, see Dai, G.-m., J. Opt. Soc. Am. A,23:539-543 (2006), Janssen, A. J. E. M., J. Microlith., Microfab.,Microsyst., 5:030501 (2006), Goldberg K. A. et al., J. Opt. Soc. Am. A,18:2146-2152 (2001), Schwiegerling, J., J. Opt. Soc. Am. A, 19:1937-1945(2002), Campbell, C. E., J. Opt. Soc. Am. A, 20:209-217 (2003), Shu, L.et al., J. Opt. Soc. Am. A, 23:1960-1968 (2006), Bará, S. et al., J.Opt. Soc. Am. A, 23:2061-2066 (2006), and Lundström L. et al., J. Opt.Soc. Am. A, 24:569-577 (2007). Zernike resizing polynomials can bewritten as

$\begin{matrix}{{{G_{n}^{i}(ɛ)} = {ɛ^{n}\sqrt{\left( {n + {2i} + 1} \right)\left( {n + 1} \right)}{\sum\limits_{j = 0}^{i}\;{\frac{\left( {- 1} \right)^{i + j}{\left( {n + i + j} \right)!}}{{j!}{\left( {n + j + 1} \right)!}{\left( {i - j} \right)!}}ɛ^{2j}}}}},} & (18)\end{matrix}$so a new set of Zernike coefficients can be related to the original setas

$\begin{matrix}{b_{n}^{m} = {\sum\limits_{i = 0}^{{({N - n})}/2}\;{{G_{n}^{i}(ɛ)}a_{n + {2i}}^{m}}}} & (19)\end{matrix}$

Table 2 shows the formulas for Zernike resizing coefficients asfunctions of the Zernike resizing polynomials. Equation (19) indicatesthat (1) the scaled Zernike coefficients may depend only upon theoriginal Zernike coefficients of the same azimuthal frequency m; (2) thescaled Zernike coefficients may not depend upon the original Zernikecoefficients of lower orders. For example, a defocus aberration may notinduce spherical aberration when the pupil constricts. On the otherhand, a spherical aberration may induce defocus aberration when thepupil constricts.

In Eq. (18), the index n is referred to as the radial order, and theindex i is referred to as the depth. When it is used for Zernikepolynomials resizing, the depth i is related to the maximum order N ofZernike polynomial expansion as i≦(N−n)/2. Table 3 shows Zernikeresizing polynomials up to the 10th order.

There are several properties concerning Zernike resizing polynomialsthat can be useful for the following discussion. (1) According to someembodiments, Zernike resizing polynomials are zero except for G₀ ⁰ whenε=1, i.e., G_(n) ^(i)(1)=0. (2) According to some embodiments, Zernikeresizing polynomials of depth zero equal the power of ε, i.e., G_(n)^(o)(ε)=ε^(n). (3) According to some embodiments, Zernike resizingpolynomials except for G_(n) ^(o) can be expressed as the difference oftwo Zernike radial polynomials as functions of ε. A detailed discussionand proof of these properties is given in Appendix A.

Table 2 shows Zernike resizing coefficients expressed as the originalZernike coefficients, where ε(<1) is the pupil resizing ratio.

TABLE 2 n New Coefficients b_(n) ^(m) 0 G₀ ⁰(ε)a₀ ⁰ + G₀ ¹(ε)a₂ ⁰ + G₀²(ε)a₄ ⁰ + G₀ ³(ε)a₆ ⁰ + G₀ ⁴(ε)a₈ ⁰ + G₀ ⁵(ε)a₁₀ ⁰ 1 G₁ ⁰(ε)a₁ ^(m) +G₁ ¹(ε)a₃ ^(m) + G₁ ²(ε)a₅ ^(m) + G₁ ³(ε)a₇ ^(m) + G₁ ⁴(ε)a₉ ^(m) 2 G₂⁰(ε)a₂ ^(m) + G₂ ¹(ε)a₄ ^(m) + G₂ ²(ε)a₆ ^(m) + G₂ ³(ε)a₈ ^(m) + G₂⁴(ε)a₁₀ ^(m)+ 3 G₃ ⁰(ε)a₃ ^(m) + G₃ ¹(ε)a₅ ^(m) + G₃ ²(ε)a₇ ^(m) + G₃³(ε)a₉ ^(m) 4 G₄ ⁰(ε)a₄ ^(m) + G₄ ¹(ε)a₆ ^(m) + G₄ ²(ε)a₈ ^(m) + G₄³(ε)a₁₀ ^(m) 5 G₅ ⁰(ε)a₅ ^(m) + G₅ ¹(ε)a₇ ^(m) + G₅ ²(ε)a₉ ^(m)+ 6 G₆⁰(ε)a₆ ^(m) + G₆ ¹(ε)a₈ ^(m) + G₆ ²(ε)a₁₀ ^(m) 7 G₇ ⁰(ε)a₇ ^(m) + G₇¹(ε)a₉ ^(m) 8 G₈ ⁰(ε)a₈ ^(m) + G₈ ¹(ε)a₁₀ ^(m) 9 G₉ ⁰(ε)a₉ ^(m) 10 G₁₀⁰(ε)a₁₀ ^(m)

According to embodiments of the present invention, it is possible toexpress the resized coefficient of the vertical coma as a function ofthe coefficients of primary, secondary, and tertiary coma. Consider thespecial case for ε=0.8, such as a 6 mm pupil constricts to 4.8 mm. FromTable 2, we have b₃ ¹=G₃ ⁰(ε)a₃ ¹+G₃ ¹(ε)a₅ ¹+G₃ ²(ε)a₇ ¹. Substitutingthe Zernike resizing polynomials from Table 3, we get b₃ ¹=ε³[a₃¹−2√{square root over (6)}(1−ε²)a₅ ¹+2√{square root over(2)}(−12ε²+7ε⁴)a₇ ¹]. Similarly, for b₃ ⁻¹, we have b₃ ⁻¹=ε³[a₃¹−2√{square root over (6)}(−ε²)a₅ ⁻¹+2√{square root over(2)}(−12ε²+7ε⁴)a₇ ⁻¹]. Because G₃ ¹(0.8)=−2√{square root over(6)}(1−0.8²)×0.8³=−0.903, G₃ ²(0.8)=2√{square root over(2)}(5−12×0.8²+7×0.8⁴)×0.8³=0.271, and G³ ₃(0.8)=−2√{square root over(10)}(5−21×0.8²+28×0.8⁴−12×0.8⁶)×0.8³=−0.379, we find b₃ ¹=0.8³a₃¹−0.903a₅ ¹+0.271a₇ ¹−0.379a₉ ¹)=0.512a₃ ¹−0.903a₅ ¹+0.27a₇ ¹+0.379a₉ ¹.Similarly, b₃ ⁻¹=0.512a₃ ⁻¹−0.903a₅ ⁻¹+0.271a₇ ⁻¹−0.379a₉ ⁻¹.

Table 3 shows Zernike resizing polynomials up to the 10th order.

TABLE 3 n i G^(i) _(n)(ε) 0 1 −{square root over (3)}(1 − ε²) 0 2{square root over (5)}(1 − 3ε² + 2ε⁴) 0 3 −{square root over (7)}(1 −6ε² + 10ε⁴ − 5ε⁶) 0 4 {square root over (9)}(1 −10ε² + 30ε⁴ − 35ε⁶ +14ε⁸) 0 5 −{square root over (11)}(1 − 15ε² + 70ε⁴ − 140ε⁶ + 126ε⁸ −42ε¹⁰) 1 1 −2{square root over (2)}ε(1 − ε²) 1 2 {square root over(3)}ε(3 − 8ε² + 5ε⁴) 1 3 −4ε(2 − 10ε² + 15ε⁴ − 7ε⁶) 1 4 {square rootover (5)}ε(5 − 40ε² + 105ε⁴ − 112ε⁶ + 42ε⁸) 2 1 −{square root over(15)}ε²(1 − ε²) 2 2 {square root over (21)}ε²(2 − 5ε² + 3ε⁴) 2 3−{square root over (3)}ε²(10 − 45ε² + 63ε⁴ − 28ε⁶) 2 4 {square root over(33)}ε² (5 − 35ε² + 84ε⁴ − 84ε⁸ + 30ε⁸) 3 1 −2{square root over (6)}ε³(1− ε²) 3 2 2{square root over (2)}ε³(5 − 12e²7ε⁴) 3 3 −2{square root over(10)}ε³(5 − 21ε² + 28ε⁴ − 12ε⁶) 4 1 −{square root over (35)}ε⁴(1 − ε²) 42 3{square root over (5)}ε⁴(3 − 7ε² + 4ε⁴) 4 3 −{square root over(55)}ε⁴(7 − 28ε² + 36ε⁴ − 15ε⁶) 5 1 −4{square root over (3)}ε⁵(1 − ε²) 52 {square root over (15)}ε⁵(1 − 16ε² + 9ε⁴) 6 1 −3{square root over(7)}ε⁶(1 − ε²) 6 2 {square root over (77)}ε⁶(4 − 9ε² + 5ε⁴) 7 1−4{square root over (5)}ε⁷(1 − ε²) 8 1 −3{square root over (11)}ε⁸(1 −ε²)

Table 4 shows a set of Zernike coefficients and the correspondingresized Zernike coefficients when a pupil resizing ratio of 0.75 isassumed. The original wavefront map 900 a shown in FIG. 9A and theresized wavefront map 900 b shown in FIG. 9B correspond to Zernikecoefficients listed in Table 4. The resized wavefront appears identicalto the inner part of the original wavefront within the new pupil size.Table 4 shows Zernike coefficients before (a_(n) ^(m)) and after (b_(n)^(m)) pupil constriction (ε=0.75).

TABLE 4 i n m a_(n) ^(m) b_(n) ^(m) 0 0 0 0.8724 0.5849 1 1 −1 −0.6983−0.5119 2 1 1 0.1979 −0.1070 3 2 −2 −0.1216 −0.2145 4 2 0 0.3600 0.11975 2 2 0.2358 0.2308 6 3 −3 0.0624 0.0140 7 3 −1 −0.0023 −0.0831 8 3 10.2665 0.1814 9 3 3 0.1608 −0.0546 10 4 −4 0.0725 −0.0324 11 4 2 0.15900.0376 12 4 0 0.0801 0.0404 13 4 2 −0.0790 −0.0781 14 4 4 −0.0841−0.0597 15 5 −5 −0.0635 −0.0151 16 5 −3 0.0136 0.0032 17 5 −1 0.09080.0215 18 5 1 −0.0763 −0.0181 19 5 3 0.1354 0.0321 20 5 5 0.0227 0.005421 6 −6 −0.0432 −0.0077 22 6 −4 0.0676 0.0120 23 6 −2 0.0155 0.0028 24 60 −0.0184 −0.0033 25 6 2 0.0649 0.0116 26 6 4 0.0404 0.0072 27 6 60.0842 0.0150

2.5 Effective Power and Correction of Presbyopia

Traditionally, the refractive power is referred to as the sphere andcylinder that best correct the refractive error of the human eye so asto achieve the best visual acuity. Therefore, the refractive power maybe independent of pupil size. When high order aberrations exist, asdiscussed in Dai, G.-m., J. Opt. Soc. Am. A, 23:539-543 (2006), therefractive power may be pupil size dependent. The instantaneousrefractive power that is dependent upon the pupil size can be termedeffective power. For an ocular wavefront that is associated with the setof Zernike coefficients {a_(i)}, when the pupil constricts the new setof Zernike coefficients becomes {b_(i)}. Aspects of effective power arediscussed in U.S. Patent Publication No. 2005/0270491.

The sphere, cylinder, and cylinder axis in the plus cylinder notationafter the pupil constriction can be written as

$\begin{matrix}{{S = {{- \frac{4\sqrt{3}b_{2}^{0}}{ɛ^{2}R^{2}}} - \frac{2\sqrt{6}\sqrt{\left( b_{2}^{- 2} \right)^{2} + \left( b_{2}^{2} \right)^{2}}}{ɛ^{2}R^{2}}}},} & \left( {20a} \right) \\{{C = \frac{4\sqrt{6}\sqrt{\left( b_{2}^{- 2} \right)^{2} + \left( b_{2}^{2} \right)^{2}}}{ɛ^{2}R^{2}}},} & \left( {20b} \right) \\{\theta = {\frac{1}{2}{{\tan^{- 1}\left( \frac{b_{2}^{- 2}}{b_{2}^{2}} \right)}.}}} & \left( {20c} \right)\end{matrix}$

From Table 2, we have

$\begin{matrix}{{b_{2}^{- 2} = {\sum\limits_{i = 0}^{{N/2} - 1}\;{{G_{2}^{i}(ɛ)}a_{2{({i + 1})}}^{- 2}}}},} & \left( {21a} \right) \\{{b_{2}^{0} = {\sum\limits_{i = 0}^{{N/2} - 1}\;{{G_{2}^{i}(ɛ)}a_{2{({i + 1})}}^{0}}}},} & \left( {21b} \right) \\{b_{2}^{2} = {\sum\limits_{i = 0}^{{N/2} - 1}\;{{G_{2}^{i}(ɛ)}{a_{2{({i + 1})}}^{2}.}}}} & \left( {21c} \right)\end{matrix}$

Substituting Eq. (21) into Eq. (20), we have

$\begin{matrix}{{S = {{{- \frac{4\sqrt{3}}{ɛ^{2}R^{2}}}{\sum\limits_{i = 0}^{{N/2} - 1}\;{{G_{2}^{i}(ɛ)}a_{2{({i + 1})}}^{0}}}} - {C/2}}},} & \left( {22a} \right) \\{C = {{- \frac{4\sqrt{6}}{ɛ^{2}R^{2}}}\left\{ {\sum\limits_{i = 0}^{{N/2} - 1}\;{\sum\limits_{i^{\prime} + 1}^{{N/2} - 1}\;{{G_{2}^{i}(ɛ)}{{G_{2}^{i^{\prime}}(ɛ)}\left\lbrack {{a_{2{({i + 1})}}^{- 2}a_{2{({i^{\prime} + 1})}}^{- 2}} + a_{2{({i + 1})}}^{2} + a_{2{({i^{\prime} + 1})}}^{2}} \right\rbrack}}}} \right\}^{1/2}}} & \left( {22b} \right) \\{\theta = {\frac{1}{2}{{\tan^{- 1}\left\lbrack \frac{\sum\limits_{i = 0}^{{N/2} - 1}\;{{G_{2}^{i}(ɛ)}a_{2{({i + 1})}}^{- 2}}}{\sum\limits_{i = 0}^{{N/2} - 1}\;{{G_{2}^{i}(ɛ)}a_{2{({i + 1})}}^{2}}} \right\rbrack}.}}} & \left( {22c} \right)\end{matrix}$

Equations (22a), (22b), and (22c) can be combined to determine arefraction for a general resizing case.

For the first four orders of Zernike polynomials, or N=4, Eq. (22) canbe written as

$\begin{matrix}{{S = {{- {\frac{4\sqrt{3}}{R^{2}}\left\lbrack {a_{2}^{0} - {\sqrt{15}\left( {1 - ɛ^{2}} \right)a_{4}^{0}}} \right\rbrack}} - {C/2}}},} & \left( {23a} \right) \\{{C = {{- \frac{4\sqrt{6}}{e^{2}R^{2}}}\left\{ {\left( a_{2}^{- 2} \right)^{2} + \left( a_{2}^{2} \right)^{2} - {2\sqrt{15}{\left( {1 - ɛ^{2}} \right)\left\lbrack {{a_{2}^{- 2}a_{4}^{- 2}} + {a_{2}^{2}a_{4}^{2}}} \right\rbrack}} + {15{\left( {1 - ɛ^{2}} \right)^{2}\left\lbrack {\left( a_{4}^{- 2} \right)^{2} + \left( a_{4}^{2} \right)^{2}} \right\rbrack}}} \right\}^{1/2}}},} & \left( {23b} \right) \\{\theta = {\frac{1}{2}{{\tan^{- 1}\left\lbrack \frac{a_{2}^{- 2} - {\sqrt{15}\left( {1 - ɛ^{2}} \right)a_{4}^{- 2}}}{a_{2}^{2} - {\sqrt{15}\left( {1 - ɛ^{2}} \right)a_{4}^{2}}} \right\rbrack}.}}} & \left( {23c} \right)\end{matrix}$

Table 5 shows Zernike coefficients before the pupil constriction for a 6mm pupil (R=3 mm)

TABLE 5 Zernike index i n m a_(n) ^(m) 3 2 −2 2.315 4 2 0 3.630 5 2 2−1.288 11 4 −2 0.075 12 4 0 −0.230 13 4 2 −0.158 23 6 −2 0.042 24 6 00.089 25 6 2 −0.012

Table 6 shows wavefront refractions over different pupil sizes.

TABLE 6 Pupil size (mm) 6 5 4 3 2 1 0 Sphere (D) −1.35 −1.65 −1.97 −2.27−2.52 −2.67 −2.73 Cylinder (D) −2.88 −2.70 −2.64 −2.66 −2.70 −2.74 −2.76Axis 59.5° 58.2° 56.8° 55.6° 54.7° 54.2° 54.0°

For the minus cylinder notation, Eqs. (22) and (23) can be modifiedaccordingly by changing the sign of C from plus to minus.

Equation (22), and Eq. (23) as a special case, indicate that thespherical equivalent (S+C/2) can depend upon defocus, primary,secondary, tertiary, and other higher order spherical aberrations of theoriginal wavefront when the pupil size constricts. Similarly, thecylinder can depend upon the primary, secondary, tertiary, and otherhigher order astigmatism of the original wavefront when the pupil sizeconstricts.

According to embodiments of the present invention, it is possible tocalculate the sphere, cylinder, and cylinder axis in the minus cylindernotation as a function of the pupil size for the Zernike coefficientsshown in Table 5.

For the cylinder, we have C=−(4√{square root over(6)}/3²){[2.315−√{square root over (15)}(1−ε²)×0.075+√{square root over(21)}(2−5ε²+3ε⁴)×0.042]²[−1.288−√{square root over(15)}(1−ε²)×(−0.158)+√{square root over(21)}(2−5ε²+3ε⁴)×(−0.012)]²}^(1/2), S=−(4√{square root over(3)}/3²)[3.630−√{square root over (15)}(1−ε²)×(−0.230)+√{square rootover (21)}(2−5ε²+3ε⁴)×0.089]−C/2. θ=tan⁻¹{[2.315−√{square root over(15)}(1−ε²)×0.075+√{square root over(21)}(2−5ε²+3ε⁴)×0.042]/[−1.288−√{square root over(15)}(1−ε²)×(−0.158)+√{square root over(21)}(2−5ε²+3ε⁴)×(−0.012)]}×90/π. Applying ε from 6 to 0, we obtain therespective values of the sphere, the cylinder, and the cylinder axis, asshown in Table 6. FIG. 10 shows effective power curves for sphere andcylinder as a function of pupil size. In some embodiments, FIG. 10 showseffective power curves of the sphere and cylinder when the values of εare continuous.

The power curves of the sphere and cylinder as a function of the pupilsize can be useful for the correction of presbyopia, as the pupil sizeconstricts during accommodation. Aspects of this feature are describedin Lowenfeld, I. E., The Pupil: Anatomy, Physiology, and ClinicalApplications, Butterworth-Heinemann, Boston (1999). A sphere power curveas shown in FIG. 10 may make the eye more myopic when the pupilconstricts.

Suppose an emmetropic subject needs a presbyopic correction so heremains emmetropic when the pupil size is 4.5 mm but becomes −1.5 Dmyopic when the pupil constricts to 2.25 mm. It is possible to determinethe amount of spherical aberration at 4.5 mm necessary to achieve that.The subject is emmetropic so he does not have cylinder error, or C=0.From Eq. (23), we obtain S=−(4√{square root over (3)}/2.252)[a₂⁰−√{square root over (15)}(1−ε²)a₄ ⁰]. When the pupil size is 4.5 mm,the subject remains emmetropic. Therefore, for ε=1, S=−(4√{square rootover (3)}/2.25²)a₂ ⁰=0, or a₂ ⁰=0. For ε=2.25/4.5=0.5, the subject wantsto achieve −1.5 D. That means S=−(4√{square root over(3)}/2.25²)(−√{square root over (15)})[1−(½)²]a₄ ⁰=−1.5, or a₄ ⁰=−0.38μm. ( Hence, if we introduce 0.38 microns of negative sphericalaberration to the eye, this subject can have a manifest refraction of−1.5 D at 2.25 mm pupil even though his manifest refraction is zero at4.5 mm.

2.6 Pupil Resizing with Seidel Series

The set of Seidel series is a set of basis functions for describingoptical aberrations. Although this set of basis functions may not bepractical in all circumstances for ocular aberration representationbecause most ocular aberrations are not symmetric, it is useful to testpupil resizing embodiments described herein. Seidel series typicallyrequire x-axis rotational symmetry, and in normal aberrations suchrotational symmetry is not observed. Nonetheless, Seidel series may beused because it represents a classical aberration.

Table 7 shows Seidel coefficients before (a_(n) ^(m)) and after (b_(n)^(m)) pupil constriction (ε=0.85).

TABLE 7 i n m a_(n) ^(m) b_(n) ^(m) 0 0 0 −0.3386 −0.3386 1 1 1 0.45010.3252 2 2 0 −0.2689 −0.1943 3 2 2 0.0712 0.0437 4 3 1 −0.0093 −0.0057 53 3 0.1956 0.1021 6 4 0 0.1310 0.0684 7 4 2 −0.0218 −0.0114 8 4 4−0.1926 −0.0855 9 5 1 0.1286 0.0571 10 5 3 −0.0221 −0.0098 11 5 5 0.03850.0145 12 6 0 0.0973 0.0367 13 6 2 0.1406 0.0530 14 6 4 0.0794 0.0299 156 6 −0.0925 −0.0297

A Seidel series can be expressed ass _(n) ^(m)(ρ,θ)=ρ^(n) cos^(m) θ.   (24)

Since often Seidel radial polynomials are exactly the same as the radialpolynomials in Taylor monomials, it can shown that the set of the pupilresizing polynomials is the same as in Taylor monomials asL _(n)(ε)=ε^(n).   (25)

Hence, similar to Taylor monomials, each new Seidel coefficient can bescaled by ε^(n) where n is the radial order of the Seidel series.Equation (25) can represent the GPRF of a Seidel series.

Table 7 shows a set of Seidel coefficients and the corresponding resizedSeidel coefficients when a pupil resizing ratio of 0.85 is assumed. Theoriginal wavefront map 1100 a shown in FIG. 11A and the resizedwavefront map 1100 b shown in FIG. 11B correspond to Seidel coefficientslisted in Table 7. The resized wavefront appears identical to the innerpart of the original wavefront within the new pupil size. As can be seenfrom FIGS. 11A and 11B, the wavefront can be symmetric with respect tothe x-axis. In some embodiments, the set of Seidel series may not beapplicable to ocular wavefront representation.

3. Wavefront Representation for Cyclorotation

The ocular wavefront representation for cyclorotation can be consideredin vision correction because human eyes have three degrees of freedom inrotational eye movements, as discussed in Chernyak, D. A., J. Cataract.Refract. Surg., 30:633-638 (2004). This is also shown in FIGS. 12A to12C, where FIG. 12A represents cyclorotation and FIGS. 12B and 12Crepresent pupil center shift. FIG. 12A shows an eye 1200 a, having apupil 1205 a, rotating about a longitudinal axis 1210 a. FIG. 12B showsan eye 1200 b, having a pupil 1205 b, rotating about a longitudinal axis1210 b. FIG. 12C shows an eye 1200 c, having a pupil 1205 c, rotatingabout a longitudinal axis 1210 c. In this section, the cyclorotation ofocular wavefront maps is discussed. The pupil center shift caused by theeye movement is discussed in the next section. It is possible to evokethe directional or linear eye tracking, aspects of which are discussedin Yee, K. “Active eye tracking for excimer laser refractive surgery,”in Aberration-Free Refractive Surgery, 2nd ed., J. Bille, C. F. H.Hamer, and F. H. Loesel, eds. (Springer, 2003), pp. 125-140, or thecyclotorsional eye tracking, aspects of which are discussed in Chernyak,D. A., IEEE Trans. Bio. Eng., 52:2032-2040 (2005), during laserrefractive surgery.

FIG. 13 shows the coordinates of a wavefront 1300 before (solid lines)and after (dashed lines) cyclorotation of the wavefront by an angle φ.In some embodiments, a counter clockwise angle can be defined aspositive. A relation between the coordinates can be given in thefollowing equation.x′=x cos φ+y sin φ,   (26a)y′=−x sin φ+y cos φ.   (26b)

3.1 Wavefront Rotation with Taylor Monomials

For the wavefront rotation with Taylor monomials, it can be shown(Appendix B) that the Taylor coefficients after the rotation are relatedto the original Taylor coefficients by

$\begin{matrix}{{b_{p}^{q} = {\sum\limits_{k = 0}^{q}\;{\sum\limits_{l = 0}^{p - q}{\frac{\left( {- 1} \right)^{k}{q!}{\left( {p - q} \right)!}}{{k!}{l!}{\left( {q - k} \right)!}{\left( {p - q - l} \right)!}}\left( {\sin\;\phi} \right)^{k + l}\left( {\cos\;\phi} \right)^{p - k - l}{a_{p}^{q - k + l}.}}}}}\;} & (27)\end{matrix}$

Table 8 lists the conversion formulas for an original set of Taylorcoefficients to a new set of Taylor coefficients when a cyclorotation ofthe wavefront map by an angle φ counter clockwise occurs. As shown here,Taylor coefficients of a rotated wavefront b_(i) can be represented as afunction of the original Taylor coefficients a_(i) for p≦5.

TABLE 8 p q Formula 0 0 b₀ = a₀ 1 0 b₁ = a₁ cos φ + a₂ sin φ 1 1 b₂ =−a₁ sin φ + a₂ cos φ 2 0 b₃ = a₃ cos² φ + 2a₄ sin φ cosφ + a₅ sin² φ 2 1b₄ = −a₃ sin φ cos φ + a₄(cos² φ − sin² φ) + a₅ sin φ cosφ 2 2 b₅ = a₃sin² φ − 2a₄ sin φ cos φ + a₅ cos² φ 3 0 b₆ = a₆ cos3 φ + 3a₇ cos² φ sinφ + 3a₈ cos φ sin² φ + a₉ sin³ φ 3 1 b₇ = −a₆ cos² φ sin φ + a₇(cos² φ −2sin² φ) cos φ + a₈(2 cos² φ − sin² φ) sin φ + a₉ cos φ sin² φ 3 2 b₈ =a₆ cos φ sin² φ − a₇(2 cos² φ − sin² φ) sin φ + a₈ (cos² φ − 2 sin² φ)cos φ + a₉ cos² φ sin φ 3 3 b₉ = −a₆ sin³ φ + 3a₇ cos φ sin² φ − 3a₈cos² φ sin φ + a₉ cos³ φ 4 0 b₁₀ = a₁₀ cos⁴ φ + 4a₁₁ cos³ φ sin φ + 6a₁₂cos² φ sin² φ + 4a₁₃ cos φ sin³ φ + a₁₄ sin⁴ φ 4 1 b₁₁ = −a₁₀ cosφ sinφ + a₁₁ cosφ (cosφ − 3sin² φ) + 3a₁₂ sin φ cos φ (cos² φ − sin² φ) + a₁₃sin² φ (3cos² φ − sin² φ) + a₁₄ sin³ φ cos φ 4 2 b₁₂ = a₁₀ sin² φ cos² φ− 2a₁₁ sin φ cos φ (cos² φ − sin² φ) + a₁₂ sin² φ (4 cos² φ + sin² φ) +2a₁₃ sin φ cos φ (cos² φ − sin² φ) + a₁₄ sin² φ cos² φ 4 3 b₁₃ = −a₁₀sin³ φ cos φ + a₁₁ sin² φ (3 cos² φ − sin² φ) − 3a₁₂ sin φ cos φ (cos² φ− sin² φ) + a₁₃ cos² φ (cos² φ − 3 sin² φ) + a₁₄ sin φ cos³ φ) 4 4 b₁₄ =a₁₀ sin⁴ φ − 4a₁₁ sin³ φ cos φ + 6a₁₂ sin² φ cos² φ − 4a₁₃ sin φ cos³φ + a₁₄ cos⁴ φ 5 0 b₁₅ = a₁₅ cos⁵ φ + 5a₁₆ sin φ cos⁴ φ + 10a₁₇ sin² φcos³ φ + 10a₁₈ sin³ φ cos² φ + 5a₁₉ sin⁴ φ cos φ + a₂₀ sin⁵ φ 5 1 b₁₆ =−a₁₅ sin φ cos⁴ φ + a₁₆ cos³ φ (cos² φ − 4 sin² φ) + 2a₁₇ sin φ cos² φ(2cos² φ − 3 sin² φ) + 2a₁₈ sin² φ cos φ (3 cos² φ − 2 sin² φ) + a₁₉sin³ φ (4 cos² φ − sin² φ) + a₂₀ sin⁴ φ cos φ 5 2 b₁₇ = a₁₅ sin² φ cos³φ − a₁₆ sin φ cos² φ (2 cos2 φ − 3 sin² φ) + a₁₇ cos φ (cos⁴ φ − 6 sin²φ cos² φ + 3 sin⁴ φ) + a₁₈ sin φ (3 cos⁴ φ − 6 sin² φ cos² φ + sin⁴ φ) +a₁₉ sin² φ cos φ (3 cos² φ − 2 sin²φ) + a₂₀ sin³ φ cos² φ 5 3 b₁₈ = a₁₅sin³ φ cos² φ + a₁₆ sin² φ cos φ (3 cos² φ − 2 sin² φ) − a₁₇ sin φ (3cos⁴ φ − 6 sin² φ cos² φ + sin⁴ φ) + a₁₈ cos φ (3 sin⁴ φ − 6 sin² φ cos²φ + cos⁴ φ) + a₁₉ sin φ cos² φ (2 cos² φ − 3 sin² φ) + a₂₀ sin² φ cos³ φ5 4 b₁₉ = a₁₅ sin⁴ φ cos φ + a₁₆ sin³ φ (sin² φ − 4 cos² φ) + 2a₁₇ sin²φ cos φ (3 cos² φ − 2 sin² φ) − 2a₁₈ sin φ cos² φ (2 cos² φ − 3 sin² φ)− a₁₉ cos³ φ (4 sin² φ − cos² φ) + a₂₀ sin φ cos⁴ φ 5 5 b₂₀ = −a₁₅ sin⁵φ + 5a₁₆ φ sin⁴ φ cos φ − 10a₁₇ sin³ φ cos² φ + 10a₁₈ sin² φ cos³ φ −5a₁₉ sin φ cos⁴ φ + a₂₀ cos⁵ φ

Because the radial order of both a_(p) ^(q−k+l) and b_(p) ^(q) is p, Eq.(27) indicates that Taylor coefficients after cyclorotation can beaffected by those in the same order. For example, b₃, b₄, and b₅ can beaffected by a₃, a₄, and a₅ because they are in the same radial order.Similarly, b₁₀ to b₁₄ can be affected by a₁₀ to a₁₄ because they arealso in the same order. Table 8 confirms this observation.

As an example, Table 9 shows a list of Taylor coefficients of anoriginal wavefront and the corresponding Taylor coefficients of thewavefront when it rotates by 90°, 180°, and 270°, respectively. Thecorresponding wavefront maps 1400 a, 1400 b, 1400 c, and 1440 d, areshown in FIGS. 14A to 14D, respectively. As can be seen from these maps,the features of the maps are rotated by the respective angles. FIG. 14Ashows the original wavefront map. FIG. 14B shows the rotated map after90° rotation. FIG. 14C shows the rotated map after 180° rotation. FIG.14D shows the rotated map after 270° rotation.

It should be noted that when the rotational angle is not a multiple of90°, error can occur when the wavefront is digitized, or sampled. Thisis because the formulas shown in Table 8 are analytical and correspondto a smooth wavefront with infinite sampling. With Taylor monomials,because of the power in the triangular functions, any error can beamplified. Therefore, according to some embodiments the set of Taylormonomials may not be ideal for the study of the wavefront rotation.Table 9 shows an example of the wavefront rotation with Taylorcoefficients for the original and the rotated wavefronts after variousrotation angles.

TABLE 9 I p q Original 90° 180° 270° 0 0 0 1.6524 1.6524 1.6524 1.6524 11 0 −1.7143 0.5963 1.7143 −0.5963 2 1 1 0.5963 1.7143 −0.5963 −1.7143 32 0 −4.0792 −1.7784 −4.0792 −1.7784 4 2 1 −6.3573 6.3573 −6.3573 6.35735 2 2 −1.7784 −4.0792 −1.7784 −4.0792 6 3 0 5.5547 −5.8774 −5.55475.8774 7 3 1 −5.2032 −1.1222 5.2032 1.1222 8 3 2 1.1222 −5.2032 −1.12225.2032 9 3 3 −5.8774 −5.5547 5.8774 5.5547 10 4 0 11.3340 4.4274 11.33404.4274 11 4 1 8.7331 −22.8555 8.7331 −22.8555 12 4 2 1.6505 1.65051.6505 1.6505 13 4 3 22.8555 −8.7331 22.8555 −8.7331 14 4 4 4.427411.3340 4.4274 11.3340 15 5 0 −3.5909 4.9062 3.5909 −4.9062 16 5 15.9912 1.2298 −5.9912 −1.2298 17 5 2 5.8266 6.2527 −5.8266 −6.2527 18 53 6.2527 −5.8266 −6.2527 5.8266 19 5 4 −1.2298 5.9912 1.2298 −5.9912 205 5 4.9062 3.5909 −4.9062 −3.5909 21 6 0 −10.3417 −3.4241 −10.3417−3.4241 22 6 1 −6.2927 17.9847 −6.2927 17.9847 23 6 2 −11.4756 −6.2223−11.4756 −6.2223 24 6 3 −21.4397 21.4397 −21.4397 21.4397 25 6 4 −6.2223−11.4756 −6.2223 −11.4756 26 6 5 −17.9847 6.2927 −17.9847 6.2927 27 6 6−3.4241 −10.3417 −3.4241 −10.3417

3.2 Wavefront Rotation with Zernike Polynomials

Many refractive laser companies use Zernike polynomial representedocular wavefronts to drive customized laser vision correction.Embodiments of the present invention provide systems and methods fordetermining new Zernike coefficients based on an original set when acyclorotation of the wavefront occurs during a vision correction ortreatment procedure. For example, a patient's ocular wavefront ismeasured with an aberrometer. However, during the refractive laserablation, the patient's eye may exhibit a cyclotorsional movement. Thetreatment or ablated shape therefore may not exactly land on thelocation as intended, but instead may be rotated by a certain angle.This would lead to a residual wavefront error that is not zero, henceaffecting the visual outcome after the correction or treatment.

From the definition of Zernike polynomials, it can be shown (Appendix C)that the new Zernike coefficients are related to the original Zernikecoefficients of the same radial degree n and the absolute value of theazimuthal frequency m asb _(n) ^(−|m|) =a _(n) ^(−|m|)cos|m|φ+a _(n) ^(m) sin|m|φ,   (28a)b _(n) ^(|m|) =−a _(n) ^(−|m|)sin|m|φ+a _(n) ^(|m|)cos|m|φ.   (28b)

Equations (28a) and (28b) represent the Zernike formulas. It isunderstood that the cylinder axis can be represented by the originalaxis offset by a rotational angle difference, and sphere and cylindercan be the same as the original. Hence, if cylinder is present androtation occurs, the magnitude of the sphere and cylinder remains thesame and the angle of cylinder axis changes. Thus, the refraction can bethe same, except the angle will be changed. When combining decentration,rotation, and constriction, then the refraction may change due to thecombination of changes, but when considering rotation only, thenrefraction may not change, except for the angle of rotation. Table 8shows the conversion formulas for calculating the new Zernikecoefficients b_(i) from the original set a_(i) when an angle φ ofrotation counter clockwise happens. Because Z₀, Z₄, Z₁₂, and Z₂₄ arerotationally symmetric, their corresponding coefficients may not changewith respect to the rotation.

Some have suggested a vector representation for Zernike polynomials.See, for example, Campbell, C. E., Optom. Vis. Sci., 80:79-83 (2003).Zernike polynomials can be written asZ _(n) ^(m)(ρ,θ;α)=√{square root over (2−δ_(m0))}

_(n) ^(|m|)(ρ)cos [m(θ−α)]  (29)where the coefficient that combines the two symmetric Zernike termsZ_(n) ^(m) and Z_(n) ^(−m) can be calculated asc _(n,m)=√{square root over ((c _(n) ^(−m))²+(c _(n) ^(m))²)}{squareroot over ((c _(n) ^(−m))²+(c _(n) ^(m))²)},   (30)and the direction of the vector α can be calculated by

$\begin{matrix}{\alpha = {\frac{1}{m}{{\tan^{- 1}\left( \frac{c_{n}^{- {m}}}{c_{n}^{m}} \right)}.}}} & (31)\end{matrix}$

With this new representation, the rotation of the wavefront map can berepresented easily. The magnitude of the coefficient c_(n,m) does notchange, but the direction of the vector α simply becomes α−φ where φ isthe angle of the wavefront rotation.

According to embodiments of the present invention, an ocular wavefrontmay contain 0.5 μm of horizontal coma and −0.25 μm of vertical coma. Ifthis ocular wavefront map is rotated by 37° clockwise, the newhorizontal and vertical coma can be determined. The horizontal coma a₃¹=a₈=0.5 and the vertical coma a₃ ⁻¹=a₇=0.25. Rotating 37° clockwisemeans 360°−37°=323° counterclockwise, or φ=323°. From Table 10 we haveb₇=a₇ cos(323°)+a₈ sin(323°)=−0.25 cos(323°)+0.5 sin(323°)=−0.501,b₈=−a₇ sin(323°)+a₈ cos(323°)=−0.25 sin(323°)+0.5 cos(323°)=0.249.Therefore, after the rotation, the horizontal coma becomes 0.249 μm andthe vertical coma becomes −0.501 μm. If we use the vectorrepresentation, the combined coma is √{square root over (a₇ ²+a₈²)}=√{square root over (0.5²+(−0.25)²)}=0.559 μm and the direction angleis a=tan⁻¹(a₇/a₈)=tan⁻¹(−0.25/0.5)=153°. After the rotation, the coma is√{square root over (b₇ ²+b₈ ²)}=√{square root over(0.249²+(−0.501)²)}=0.559 μm and the direction angle isa=tan⁻¹(b₇/b₈)=tan⁻¹(−0.501/0.249)=116°. The new angle a is 37° lessthan the original angle, meaning that the map is rotated by 37°clockwise. Table 10 shows Zernike coefficients of the rotated wavefrontb_(i) as a function of the original Zernike coefficients a_(i) for n≦7.

TABLE 10 n m Formula 0 0 b₀ = a₀ 1 −1 b₁ = a₁ cos φ + a₂ sin φ 1 1 b₂ =−a₁ sin φ + a₂ cos φ 2 −2 b₃ = a₃ cos 2φ + a₅ sin 2φ 2 0 b₄ = a₄ 2 2 b₅= −a₃ sin 2φ + a₅ cos 2φ 3 −3 b₆ = a₆ cos 3φ + a₉ sin 3φ 3 −1 b₇ = a₇cos φ + a₈ sin φ 3 1 b₈ = −a₇ sin φ + a₈ cos φ 3 3 b₉ = −a₆ sin 3φ + a₉cos 3φ 4 −4 b₁₀ = a₁₀ cos 4φ + a₁₄ sin 4φ 4 −2 b₁₁ = a₁₁ cos 2φ + a₁₃sin 2φ 4 0 b₁₂ = a₁₂ 4 2 b₁₃ = −a₁₁ sin 2φ + a₁₃ cos 2φ 4 4 b₁₄ = −a₁₀sin 4φ + a₁₄ cos 4φ 5 −5 b₁₅ = a₁₅ cos 5φ + a₂₀ sin 5φ 5 −3 b₁₆ = a₁₆cos 3φ + a₁₉ sin 3φ 5 −1 b₁₇ = a₁₇cos φ + a₁₈ sinφ 5 1 b₁₈ = −a₁₇ sinφ + a₁₈ cos φ 5 3 b₁₉ = −a₁₆ sin 3φ + a₁₉ cos 3φ 5 5 b₂₀ = −a₁₅ sin 5φ +a₂₀ cos 5φ 6 −6 b₂₁ = a₂₁ cos 6φ + a₂₇ sin 6φ 6 −4 b₂₂ = a₂₂ cos 4φ +a₂₆ sin 4φ 6 −2 b₂₃ = a₂₃ cos 2φ + a₂₅ sin 2φ 6 0 b₂₄ = a₂₄ 6 2 b₂₅ =−a₂₃ sin 2φ + a₂₅ cos 2φ 6 4 b₂₆ = −a₂₂ sin 4φ + a₂₆ cos 4φ 6 6 b₂₇ =−a₂₁ sin 6φ + a₂₇ cos 6φ 7 −7 b₂₈ = a₂₈ cos 7φ + a₃₅ sin 7φ 7 −5 b₂₉ =a₂₉ cos 5φ + a₃₄ sin 5φ 7 −3 b₃₀ = a₃₀ cos 3φ + a₃₃ sin 3φ 7 −1 b₃₁ =a₃₁ cos φ + a₃₂ sin φ 7 1 b₃₂ = −a₃₁ sin φ + a₃₂ cos φ 7 3 b₃₃ = −a₃₀sin 3φ + a₃₃ cos 3φ 7 5 b₃₄ = −a₂₉ sin 5φ + a₃₄ cos 5φ 7 7 b₃₅ = −a₂₈sin 7φ + a₃₅ cos 7φ

As an example, FIGS. 15A to 15H show an ocular wavefront and the effectof the partial correction resulting from the cyclorotation of the eyeduring, e.g., a refractive laser surgery. Accordingly, these figures canillustrate an example for the wavefront rotation and its influence onvision correction or treatment. FIG. 15A shows an original wavefrontmap. If the wavefront 1500 a is rotated by 3°, 12°, and 47°,respectively, the corresponding maps, 1550 b, 1500 c, and 1500 d, areshown in FIGS. 15B, 15C, and 15D, respectively. If a cyclorotation ofthe eye occurs as in FIGS. 15A, 15B, 15C, and 15D while the visioncorrection or treatment is applied, the residual wavefront or the ocularaberrations that would leave without correction, is shown in FIGS. 15E,15F, 15G, and 15H, respectively. Put differently, FIGS. 15A, 15B, 15C,and 15D illustrate wavefront contour maps for 0°, 3°, 12°, and 47°rotation, respectively, and FIGS. 15E, 15F, 15G, and 15H illustrateresidual wavefront contour maps 1500 e, 1500 f, 1500 g, and 1500 h, for0°, 3°, 12°, and 47° rotation, respectively, during vision treatment orcorrection. Corresponding Zernike coefficients are listed in Table 11.

Table 11 shows Zernike coefficients for the rotated wavefronts and forthe residual wavefronts after a partial vision correction due to acyclorotation of the eye, as shown in FIGS. 15A to 15H. The originalwavefront without rotation corresponds to a typical moderate hyperopiceye with a 6 mm pupil. The residual RMS wavefront error as well as theresidual high order RMS wavefront error for the partial correction arealso shown. Note that the coefficients of all rotationally symmetricterms, such as a₀, a₄, a₁₂, and a₂₄, typically do not change after therotation.

TABLE 11 Rotated wavefronts Residual wavefronts i n m 0° 3° 12° 47° 3°12° 47° 0 0 0 0.1734 0.1734 0.1734 0.1734 0.0000 0.0000 0.0000 1 1 −10.9003 0.8709 0.7688 0.2208 −0.0294 −0.1021 −0.5480 2 1 1 −0.5377−0.5841 −0.7131 −1.0251 −0.0464 −0.1290 −0.3120 3 2 −2 1.1068 1.14161.1703 0.3131 0.0348 0.0287 −0.8572 4 2 0 −3.0140 −3.0140 −3.0140−3.0140 0.0000 0.0000 0.0000 5 2 2 0.3913 0.2735 −0.0927 −1.1314 −0.1178−0.3662 −1.0387 6 3 −3 0.1747 0.2061 0.2673 −0.0009 0.0314 0.0612−0.2682 7 3 −1 −0.0290 −0.0458 −0.0951 −0.2545 −0.0168 −0.0493 −0.1594 83 1 −0.3210 −0.3190 −0.3080 −0.1977 0.0020 0.0110 0.1103 9 3 3 0.21430.1843 0.0707 −0.2765 −0.0300 −0.1136 −0.3472 10 4 −4 −0.0276 −0.00220.0700 0.0108 0.0254 0.0722 −0.0592 11 4 −2 0.0577 0.0794 0.1385 0.20640.0217 0.0591 0.0679 12 4 0 0.1460 0.1460 0.1460 0.1460 0.0000 0.00000.0000 13 4 2 0.2109 0.2037 0.1692 −0.0723 −0.0072 −0.0345 −0.2415 14 44 0.1191 0.1222 0.1002 −0.1218 0.0031 −0.0220 −0.2220 15 5 −5 −0.1295−0.0843 0.0716 −0.0547 0.0452 0.1559 −0.1263 16 6 −3 −0.0377 −0.0429−0.0516 0.0067 −0.0052 −0.0087 0.0583 17 5 −1 0.1742 0.1827 0.20510.2408 0.0085 0.0224 0.0357 18 5 1 0.1668 0.1575 0.1269 −0.0136 −0.0093−0.0306 −0.1405 19 5 3 −0.0359 −0.0296 −0.0069 0.0516 0.0063 0.02270.0585 20 5 5 0.1575 0.1857 0.1909 −0.1964 0.0282 0.0052 −0.3873 21 6 −6−0.1474 −0.1712 −0.1410 0.0676 −0.0238 0.0302 0.2086 22 6 −4 −0.0490−0.0685 −0.1064 0.0623 −0.0195 −0.0379 0.1687 23 6 −2 0.1044 0.09120.0464 −0.1274 −0.0132 −0.0448 −0.1738 24 6 0 −0.1634 −0.1634 −0.1634−0.1634 0.0000 0.0000 0.0000 25 6 2 −0.1204 −0.1307 −0.1525 −0.0957−0.0103 −0.0218 0.0568 26 6 4 −0.0991 −0.0867 −0.0299 0.0913 0.01240.0568 0.1212 27 6 6 −0.1004 −0.0499 0.1092 −0.1651 0.0505 0.1591−0.2743 Residual RMS wavefront error 0.1687 0.5013 1.7165 Residual highorder RMS wavefront error 0.1017 0.2989 0.8573

Table 11 shows the Zernike coefficients of the original wavefront aswell as the coefficients of the rotated wavefronts with differentrotation angles. Also shown are the coefficients of the residualwavefronts assuming a partial correction of the original wavefront dueto a cyclorotation the of the eye. To estimate how much error may inducedue to the cyclorotation of the eye, the residual RMS wavefront error aswell as the residual high order RMS wavefront error for each of therotation angles is shown. For this typical eye, a rotation of 12°induces about the same amount of high order aberrations as a typicalwavefront-driven refractive surgery.

To further demonstrate the visual influence of the error due tocyclorotation of the eye during the vision correction, FIGS. 16A to 16Gshow the point spread function of the residual wavefronts due to thepartial correction and the residual wavefronts with high orderaberrations. We assume in this case that the low order aberrations canbe corrected with, e.g., a trial lens, so as to estimate the influenceof the cyclorotation on the best corrected visual acuity. Thecorresponding simulated blurred 20/20 letter E images are also shown.The top row illustrates the point spread function and the bottom rowillustrates the corresponding blurred 20/20 letter E for the wavefrontmaps shown in FIGS. 15A to 15H. Hence, FIGS. 16A to 16D refer to theresidual wavefronts as shown in FIGS. 15E to 15H. Relatedly, FIGS. 16Eto 16G refer to the residual wavefronts excluding the low orders for thewavefronts corresponding to FIGS. 15F to 15H. The field of view for allthe images of 6′×6′. The Strehl ratios from FIGS. 15A to 15H are 1,0.251, 0.045, 0.006, 0.449, 0.105, and 0.009, respectively.

4 Wavefront Representation for Decentration

As discussed in Section 3 above, rotational eye movement can cause bothcyclorotation and decentration of ocular wavefront maps. In thissection, the representation of the decentration of ocular wavefronts andits effect on the visual outcomes is discussed. Some have proposedapproximation techniques for decentration. For example, some haveproposed approximations to the first order of a Taylor expansion, forinstances where translation is minimal. Embodiments of the presentinvention provide precise determinations of decentration, regardless ofthe amount of shift or decentration present.

4.1 Wavefront Extrapolation.

When the pupil moves, some part of the known wavefront can move out ofthe pupil, and some part of the wavefront can move into the pupil.However, the part of the wavefront that moves into the pupil can beoriginally unknown because it may not be defined. This leaves us withone certain solution that a smaller pupil is used so that after thedecentration of the constricted pupil, it is still within the originalpupil.

However, this may be impractical. It is known the eye can move in threedegrees of freedom, so the pupil moves with respect to the ocularaberrations. During the move, the pupil may not constrict. But if thepupil size does not change, the part of the wavefront that moves intothe pupil is unknown. To solve this problem, it is possible extrapolatethe original wavefront to a bigger pupil size to allow for thedecentration of the pupil.

As discussed in Section 2 above, the coefficients of a set of basisfunctions can be calculated from an original set when the pupilconstricts. The same formula can be used to calculate the coefficientsof basis functions when the pupil dilates. When the number of terms inthe wavefront expansion is the same when the pupil dilates, there is aset of coefficients associated with the dilated pupil size that when thepupil constricts to the original size, the new set of coefficientsbecomes the original set.

FIGS. 17A to 17C show an example of an original ocular wavefront 1700 a,extrapolated to an ocular wavefront 1700 b corresponding to a largerpupil size, and then an ocular wavefront 1700 c corresponding to a pupilconstricted to the original pupil size. The original wavefront 1700 aand the final wavefront 1700 c are identical.

According to some embodiments of the present invention, care should betaken for the wavefront extrapolation in the following considerations.First of all, when the pupil size dilates, there might be higher spatialfrequency information that should be captured, and hence it is possibleto use more coefficients of the basis functions in the wavefrontexpansion. Once the number of basis functions increase, the aboveassumption may no longer be true, and the extrapolation can generateerror. Secondly, in practice, the coefficients of basis functions duringthe wavefront reconstruction can be related to error in theaberrometers, such as the spot detection algorithm, centroid calculationalgorithm, and the reconstruction algorithm. When the pupil dilates,such error in some cases may not scale linearly. Therefore, theextrapolation of the ocular wavefront may induce additional errorrelated to the difference in the reconstruction error with differentpupil sizes. Nevertheless, the ocular wavefront extrapolation canprovide a very useful tool in the analysis of wavefront decentration, asdiscussed in the following subsections.

4.2 Wavefront Decentration with Taylor Monomials

Because of the simple form, the set of Taylor monomials can be a usefulset in wavefront decentration analysis. Suppose the wavefront radius isR and the wavefront moves by Δx and Δy in the x- and y-directions,respectively. Because we normally use normalized coordinates, letΔu=Δx/R and Δv=Δy/R. It can be shown (Appendix D) that Taylorcoefficients of the decentered wavefront is related to Taylorcoefficients of the original wavefront by

$\begin{matrix}{{b_{i} = {\sum\limits_{i^{\prime} = 0}^{J}\;{C_{{ii}^{\prime}}^{t4t}a_{i^{\prime}}}}},} & (32)\end{matrix}$where the conversion matrix

$\begin{matrix}{{C_{{ii}^{\prime}}^{t4t} = {\sum\limits_{i^{\prime} = 0}^{J}\;{\frac{\left( {- 1} \right)^{p^{\prime} - p}{\left( q^{\prime} \right)!}{\left( {p^{\prime} - q^{\prime}} \right)!}}{{\left( {q^{\prime} - q} \right)!}{\left( {p^{\prime} - p - q^{\prime} + q} \right)!}{q!}{\left( {p - q} \right)!}}\left( {\Delta\; u} \right)^{q^{\prime} - q}\left( {\Delta\; v} \right)^{p^{\prime} - p - q^{\prime} + q}}}},} & (33)\end{matrix}$where p′≧p, q′≧q and p′−p≧q′−q, p and q are associated with the index iand p′ and q′ are associated with the index i′. The relationship betweenthe double index p, q and the single index i for Taylor monomials isgiven by Eqs (33.1) and (33.2), respectively. Eq. (33.1) shows aconversion of a single-index to a double index.{p=int[(√{square root over (8i+1)}−1)/2], q=2i−p ² −p}  (33.1)Eq. (33.2) shows a conversion of a double-index to a single-index.

$\begin{matrix}{i = {\frac{p\left( {p + 1} \right)}{2} + q}} & (33.2)\end{matrix}$

Analytical formulas for p≦6 for the decentration of ocular wavefrontsrepresented by Taylor monomials are listed in Table 12. In practice, asdiscussed in the previous subsection, these formulas can be useddirectly with the understanding that when a decentration occurs, thepart of the wavefront that moves into the pupil can be extrapolated.FIGS. 18A to 18C show an example of the original wavefront 1800 a,extrapolated to a larger pupil size 1800 b that shows both the originalwavefront 1800 b′ (solid circle) and the decentered wavefront 1800 b″(dotted circle), and the decentered wavefront 1800 c calculated directlyfrom the formulas listed in Table 12. In this example of wavefrontdecentration, FIG. 18A shows the original wavefront 1800 a with a 6 mmpupil, FIG. 18B shows the extrapolated wavefront 1800 b to 7.5 mm pupil,and FIG. 18C shows the decentered wavefront 1800 c (Δu=−0.1 andΔv=0.15). Note the lower right corner of the decentered wavefront comesfrom the extrapolated wavefront. Apparently, the calculated decenteredwavefront does represent the decentered wavefront of the extrapolatedwavefront. The corresponding Taylor coefficients are shown in Table 13.In some embodiments, this approach can be used to determine anextrapolated wavefront for a patient who desires a vision treatment forspecific viewing conditions. For example, if the wavefront is capturedwhen the patient has a first geometrical configuration with a pupil sizeof 6 mm, and the patient desires a vision treatment for viewingconditions in dim light that correspond to a second geometricalconfiguration with a pupil size of 7 mm, it is possible to extrapolatethe examined wavefront as described above, and to develop a visiontreatment based on the extrapolation. Hence, a first set of basisfunction coefficients can be determined for the evaluation context orenvironment, and a second set of basis function coefficients can bedetermined for the viewing context or environment, where the second setof coefficients is based on the first geometrical configuration, thesecond geometrical configuration, and the first set of coefficients.Similarly, a first wavefront map can be determined for the evaluationcontext or environment, and a second wavefront map can be determined forthe viewing context or environment, where the second wavefront map isbased on the first geometrical configuration, the second geometricalconfiguration, and the first wavefront map. A prescription for treatingthe patient can be based on the second set of coefficients or the secondwavefront map, for example.

4.3 Wavefront Decentration with Zernike Polynomials

Aspects of decentration of wavefronts represented by Zernike polynomialshas been discussed in, for example, Bará, S. et al., Appl. Opt.,39:3413-3420 (2000), Guirao, A. et al., J. Opt. Soc. Am. A, 18:1003-1015(2001), Bará, S. et al., J. Opt. Soc. Am. A, 23:2061-2066 (2006), andLundström L. et al., J. Opt. Soc. Am. A, 24:569-577 (2007). Ananalytical approach using Taylor expansion was suggested in Guirao, A.et al., J. Opt. Soc. Am. A, 18:1003-1015 (2001) for the calculation ofthe Zernike coefficients of a decentered wavefront from the original setof Zernike coefficients. A first order approximation was taken forpractical applications. Lundström L. et al., J. Opt. Soc. Am. A,24:569-577 (2007) reported another analytical approach with a matrixmethod that is based on a previous approach suggested in Campbell, C.E., J. Opt. Soc. Am. A, 20:209-217 (2003).

Table 12 shows decentered Taylor coefficients b_(i) as a function of theoriginal Taylor coefficients a_(i) for n≦6.

TABLE 12 p Q Formula 0 0b₀ = a₀ − a₁Δu − a₂Δu + a₃(Δv)² + a₄ΔuΔv + a₅(Δu)² − a₆(Δv)³ − a₇Δu(Δv)² − a₈(Δu)²Δv − a₉(Δu)² + a₁₀(Δv)⁴ + a₁₁Δu(Δv)³ + a₁₂(Δu)² + a₁₃(Δu)³Δv + a₁₄(Δu)⁴ − a₁₅(Δv)⁵ − a₁₆Δu(Δv)⁴ − a₁₇(Δu)²(Δv)³ − a₁₈(Δu)³(Δv)² − a₁₉(Δu)⁴Δv − a₂₀(Δv)⁵ + a₂₁(Δv)⁶ + a₂₂Δu(Δv)⁵ + a₂₃(Δu)²(Δv)⁴ + a₂₄(Δu)³(Δv)³ + a₂₅(Δu)⁴(Δv)² + a₂₆(Δu)⁵Δv + a₂₇(Δu)⁶1 0b₁ = a₁ − 2a₃Δv − a₄Δu + 3a₆(Δv)² + 2a₇ΔuΔv + a₈(Δu)² − 4a₁₀(Δv)³ − 3a₁₁Δu(Δv)² − 2a₁₂(Δu)²Δv − a₁₃(Δu)³ + 5a₁₅(Δv)⁴ + 4a₁₆Δu(Δv)³ + 3a₁₇(Δu)²(Δv)² + 2a₁₈(Δu)³Δv + a₁₉(Δu)⁴ − 6a₂₁(Δv)⁵ − 5a₂₂Δu(Δv)⁴ − 4a₂₃(Δu)²(Δv)³ − 3a₂₄(Δu)³(Δv)² − 2a₂₅(Δu)⁴Δv − a₂₆(Δv)⁵1 1b₂ = a₂ − a₄Δv − 2a₅Δu + a₇(Δv)² + 2a₈ΔuΔv + 3a₉(Δu)² − a₁₁(Δv)³ − 2a₁₂Δu(Δv)² − 3a₁₃(Δu)²Δv − 4a₁₄(Δu)³ + a₁₆(Δv)⁴ + 2a₁₇Δu(Δv)³ + 3a₁₈(Δu)²(Δv)² + 4a₁₉(Δu)³Δv + 5a₂₀(Δu)⁴ − a₂₂(Δv)⁵ − 2a₂₃Δu(Δv)⁴ − 3a₂₄(Δu)²(Δv)³ − 4a₂₅(Δu)³(Δv)² − 5a₂₆(Δu)⁴Δv − 6a₂₇(Δv)⁵2 0b₃ = a₃ − 3a₆Δv − a₇Δu + 6a₁₀(Δv)² + 3a₁₁ΔuΔv + a₁₂(Δu)² − 10a₁₅(Δv)³ − 6a₁₆Δu(Δv)² − 3a₁₇(Δu)²Δv − a₁₈(Δu)³ + 15a₂₁(Δv)⁴ + 10a₂₂Δu(Δv)³ + 6a₂₃(Δu)²(Δv)² + 3a₂₄(Δu)³Δv + a₂₅(Δu)⁴2 1b₄ = a₄ − 2a₇Δv − 2a₈Δu + 3a₁₁(Δv)² + 4a₁₂ΔuΔv + 3a₁₃(Δu)² − 4a₁₆(Δv)³ − 6a₁₇Δu(Δv)² − 6a₁₈(Δu)²Δv − 4a₁₉(Δu)³ + 5a₂₂(Δv)⁴ + 8a₂₃Δu(Δv)³ + 9a₂₄(Δu)²(Δv)2 + 8a₂₅(Δu)³Δv + 5a₂₆(Δu)⁴2 2b₅ = a₅ − a₈Δv − 3a₉Δu + a₁₂(Δv)² + 3a₁₃ΔuΔv + 6a₁₄(Δu)² − a₁₇(Δv)³ − 3a₁₈Δu(Δv)² − 6a₁₉(Δu)²Δv − 10a₂₀(Δu)³ + a₂₃(Δv)⁴ + 3a₂₄Δu(Δv)³ + 6a₂₅(Δu)²(Δv)² + 10a₂₆(Δu)³Δv + 15a₂₇(Δu)⁴3 0b₆ = a₆ − 4a₁₀Δv − a₁₁Δu + 10a₁₅(Δv)² + 4a₁₆ΔuΔv + a₁₇(Δu)² − 20a₂₁(Δv)³ − 10a₂₂Δu(Δv)² − 4a₂₃(Δu)²Δv − a₂₄(Δu)³3 1b₇ = a₇ − 3a₁₁Δv − 2a₁₂Δu + 6a₁₆(Δv)² + 6a₁₇ΔuΔv + 3a₁₈(Δu)² − 10a₂₂(Δv)³ − 12a₂₃Δu(Δv)² − 9a₂₄(Δu)²Δv − 4a₂₅(Δu)³3 2b₈ = a₈ − 2a₁₂Δv − 3a₁₃Δu + 3a₁₇(Δv)² + 6a₁₈ΔuΔv + 6a₁₉(Δu)² − 4a₂₃(Δv)³ − 9a₂₄Δu(Δv)² − 12a₂₅(Δu)²Δv − 10a₂₆(Δu)³3 3b₉ = a₉ − a₁₃Δv − 4a₁₄Δu + a₁₈(Δv)² + 4a₁₉ΔuΔv + 10a₂₀(Δu)² − a₂₄(Δv)³ − 4a₂₅Δu(Δv)² − 10a₂₆(Δu)²Δv − 20a₂₇(Δu)³4 0 b₁₀ = a₁₀ − 5a₁₅Δv − a₁₆Δu + 15a₂₁(Δv)² + 5a₂₂ΔuΔv + a₂₃(Δu)² 4 1b₁₁ = a₁₁ − 4a₁₆Δv − 2a₁₇Δu + 10a₂₂(Δv)² + 8a₂₃ΔuΔv + 3a₂₄(Δu)² 4 2 b₁₂= a₁₂ − 3a₁₇Δv − 3a₁₈Δu + 6a₂₃(Δv)² + 9a₂₄ΔuΔv + 6a₂₅(Δu)² 4 3 b₁₃ = a₁₃− 2a₁₈Δv − 4a₁₉Δu + 3a₂₄(Δv)² + 8a₂₅ΔuΔv + 10a₂₆(Δu)² 4 4 b₁₄ = a₁₄ −a₁₉Δv − 5a₂₀Δu + a₂₅(Δv)² + 5a₂₆ΔuΔv + 15a₂₇(Δu)² 5 0 b₁₅ = a15 − 6a₂₁Δv− a₂₂Δu 5 1 b₁₆ = a₁₆ − 5a₂₂Δv − 2a₂₃Δu 5 2 b₁₇ = a₁₇ − 4a₂₃Δv − 3a₂₄Δu5 3 b₁₈ = a₁₈ − 3a₂₄Δv − 4a₂₅Δu 5 4 b₁₉ = a₁₉ − 2a₂₅Δv − 5a₂₆Δu 5 5 b₂₀= a₂₀ − a₂₆Δv − 6a₂₇Δu 6 0 b₂₁ = a₂₁ 6 1 b₂₂ = a₂₂ 6 2 b₂₃ = a₂₃ 6 3 b₂₄= a₂₄ 6 4 b₂₅ = a₂₅ 6 5 b₂₆ = a₂₆ 6 6 b₂₇ = a₂₇

This section discusses a relationship between the new set of Zernikecoefficients from the original set when the wavefront is decentered. Thestrategy is to convert the original set of Zernike coefficients toTaylor coefficients, calculate a new set of Taylor coefficients from theformulas given in Table 12, and convert the new set of Taylorcoefficients to the new set of Zernike coefficients. Hence, we have

$\begin{matrix}{{b_{i} = {\sum\limits_{i^{\prime} = 0}^{J}{C_{{ii}^{\prime}}^{z\; 4z}a_{i^{\prime}}}}},} & (34)\end{matrix}$where the conversion matrix C_(ii′) ^(z4z) CAP can be calculated asC^(z4z)=C^(t2z)C^(t4t)C^(z2t),   (35)where the matrices C^(t2z) is the matrix converting Taylor coefficientsto Zernike coefficients and C^(z2t) is the matrix converting Zernikecoefficients to Taylor coefficients. Aspects of these matrices arediscussed in Dai, G.-m., “Wavefront expansion basis functions and theirrelationships” Journal of the Optical Society of America A, 23,1657-1668 (2006). Note that C^(z2t)=(C^(t2z))⁻¹ so Eq. (35) may bewritten as:C ^(z4z) =C ^(t2z) C ^(t4t)(C ^(t2z))⁻¹.   (36)

Eq. (34) provides a generic formula that, for example, can be expandedfor b₃, b₄, and b₅ so as to correspond to Table 15. Hence, Eq. (34) canprovide a full formula that can be used to calculate all terms. Table 15corresponds to three terms associated with refraction changes. The threeterms potentially effect or influence the calculation of refractions.

Table 13 provides a list of Taylor coefficients corresponding towavefronts shown in FIGS. 18A to 18C.

TABLE 13 i p q Original Extrapolated Decentered 0 0 0 0.6485 0.64850.2619 1 1 0 2.2684 2.8355 1.4310 2 1 1 −1.0775 −1.6836 −0.6566 3 2 0−2.1462 −3.3534 0.3992 4 2 1 −8.5492 −13.3581 −11.7601 5 2 2 −7.1252−13.9164 −4.3075 6 3 0 −5.7467 −11.2240 −4.3855 7 3 1 8.2492 16.1117−4.2935 8 3 2 −14.0384 −27.4187 −11.1315 9 3 3 0.7262 1.7729 4.9569 10 40 4.1616 10.1602 2.4427 11 4 1 30.0251 73.3035 32.8528 12 4 2 13.290832.4482 8.5095 13 4 3 17.8017 43.4612 21.9792 14 4 4 19.2824 58.845216.4569 15 5 0 2.1909 6.6861 2.1715 16 5 1 −10.0422 −30.6464 8.3827 17 52 15.7452 48.0505 7.5533 18 5 3 −2.2420 −6.8420 5.3144 19 5 4 11.812136.0477 11.3796 20 5 5 0.7991 3.0483 −5.1434 21 6 0 −2.7227 −10.3863−2.7227 22 6 1 −24.6981 −94.2158 −24.6981 23 6 2 −0.4933 −1.8818 −0.493324 6 3 −28.2930 −107.9292 −28.2930 25 6 4 −12.9387 −49.3572 −12.9387 266 5 −8.6282 −32.9140 −8.6282 27 6 6 −12.0612 −57.5123 −12.0612

FIGS. 19A to 19H show an example of an ocular wavefront of 6 mm indiameter and the decentered wavefronts for decentration of 0.05 mm, 0.15mm, and 0.5 mm, respectively. The corresponding residual wavefronts arealso shown if a vision correction is applied to the original wavefront.Table 14 shows the corresponding Zernike coefficients. In this exampleof wavefront decentration, FIG. 19A shows the original wavefront 1900 awith a pupil size of 6 mm, FIG. 19B shows the decentered wavefront 1900b after 0.05 mm decentration in the x direction, FIG. 19C shows thedecentered wavefront 1900 c after 0.15 mm decentration in the xdirection, and FIG. 19D shows the decentered wavefront 1900 d after 0.5mm decentration in the x direction. The residual wavefronts 1900 e, 1900f, 1900 g, and 1900 h, that correspond to wavefronts from FIGS. 19A to19D are shown in FIGS. 19E to 19H, respectively. The wavefront maps usethe same scale. To see the influence of the decentration on the visualperformance, FIGS. 20A to 20G show the point spread functions and thecorresponding simulated blurred 20/20 letter E images. The top rowillustrates the point spread function and the bottom row illustrates thecorresponding blurred 20/20 letter E for the wavefront maps shown inFIGS. 19A to 19H. Hence, FIGS. 20A to 20D refer to the residualwavefronts as shown in FIGS. 19E to 19H. Relatedly, FIGS. 20E to 20Grefer to the residual wavefronts excluding the low orders for thewavefronts corresponding to FIGS. 19F to 19H. The field of view for allthe images of 6′×6′. The Strehl ratios from FIGS. 19A to 19H are 1,0.720, 0.138, 0.025, 0.754, 0.182, and 0.020, respectively.

Table 14 lists Zernike coefficients for the decentered wavefronts andfor the residual wavefronts after a partial vision correction due to adecentration of the eye, as shown in FIGS. 19A to 19H. The originalwavefront without decentration corresponds to a low myopic eye with alot of high order aberrations with a 6 mm pupil. The residual RMSwavefront error as well as the residual high order RMS wavefront errorfor the partial correction are also shown. Note that the coefficients ofthe sixth order, i.e., a₂₁ through a₂₇ may not change after thedecentration.

TABLE 14 Decentered (mm) wavefronts Residual wavefronts i n m 0 0.050.15 0.5 0.05 0.15 0.5 0 0 0 0.4501 0.4416 0.4230 0.3130 −0.0085 −0.0271−0.1371 1 1 −1 −0.2689 −0.2651 −0.2523 −0.1310 0.0038 0.0166 0.1379 2 11 0.0712 0.0484 0.0100 0.0321 −0.0228 −0.0612 −0.0391 3 2 −2 −0.0093−0.0149 −0.0309 −0.1712 −0.0056 −0.0216 −0.1619 4 2 0 0.2609 0.24970.2207 0.0280 −0.0112 −0.0402 −0.2329 5 2 2 0.1310 0.1140 0.0682 −0.2434−0.0170 −0.0628 −0.3744 6 3 −3 −0.0218 −0.0281 −0.0321 0.0603 −0.0063−0.0103 0.0821 7 3 −1 −0.2407 −0.2361 −0.2204 −0.0843 0.0046 0.02030.1564 8 3 1 0.1607 0.1564 0.1512 0.2353 −0.0043 −0.0095 0.0746 9 3 3−0.0221 −0.0056 0.0313 0.2518 0.0165 0.0534 0.2739 10 4 −4 0.0462 0.03580.0149 −0.0592 −0.0104 −0.0313 −0.1054 11 4 −2 0.1168 0.0899 0.0305−0.2366 −0.0269 −0.0863 −0.3534 12 4 0 0.1687 0.1710 0.1658 0.04640.0023 −0.0029 −0.1223 13 4 2 0.0953 0.0841 0.0497 −0.1953 −0.0112−0.0456 −0.2906 14 4 4 −0.1079 −0.1095 −0.1195 −0.2264 −0.0016 −0.0116−0.1185 15 5 −5 −0.0314 −0.0365 −0.0468 −0.0827 −0.0051 −0.0154 −0.051316 5 −3 0.1452 0.1507 0.1616 0.1997 0.0055 0.0164 0.0545 17 5 −1 0.13900.1541 0.1844 0.2902 0.0151 0.0454 0.1512 18 5 1 −0.0299 −0.0050 0.04490.2194 0.0249 0.0748 0.2493 19 5 3 0.1312 0.1497 0.1866 0.3159 0.01850.0554 0.1847 20 5 5 −0.1263 −0.1198 −0.1068 −0.0613 0.0065 0.01950.0650 21 6 −6 −0.0420 −0.0420 −0.0420 −0.0420 0.0000 0.0000 0.0000 22 6−4 0.0895 0.0895 0.0895 0.0895 0.0000 0.0000 0.0000 23 6 −2 −0.1400−0.1400 −0.1400 −0.1400 0.0000 0.0000 0.0000 24 6 0 −0.1032 −0.1032−0.1032 −0.1032 0.0000 0.0000 0.0000 25 6 2 −0.0849 −0.0849 −0.0849−0.0849 0.0000 0.0000 0.0000 26 6 4 −0.0861 −0.0861 −0.0861 −0.08610.0000 0.0000 0.0000 27 6 6 0.0259 0.0259 0.0259 0.0259 0.0000 0.00000.0000 Residual RMS wavefront error 0.0605 0.1911 0.8661 Residual highorder RMS wavefront error 0.0510 0.1604 0.7001

The elements of the matrix C^(z4z) and the individual formulas from Eq.(34) can be complicated. A Matlab code is given in Appendix E thataccounts for the conversion of Zernike coefficients for the pupil sizechange and the wavefront rotation and decentration. For a few specialterms, namely, the sphere and cylinder, coma and trefoil, and sphericalaberration, they are discussed in detail.

The sphere and cylinder will be discussed separately in the nextsubsection. In the following, certain high order aberration arediscussed, namely the spherical aberration. Using Eq. (34), it can beshown thatb ₁₂ =a ₁₂−2√{square root over (15)}(a ₁₇ Δv+a ₁₈ Δu)+3√{square rootover (35)}(2a ₂₄+√{square root over (2)}a ₂₅)(Δu)²+6√{square root over(70)}a ₂₃ ΔuΔv+3√{square root over (35)}(2a ₂₄−√{square root over (2)}a₂₅)(Δv)².   (37)

Equation (37) indicates that the secondary coma (Z₁₇ and Z₁₈), thetertiary astigmatism (Z₂₃ and Z₂₅), and the secondary sphericalaberration (Z₂₄) induce the primary spherical aberration (Z₁₂) when anocular wavefront is decentered.

Another high order aberration is the coma. From Eq. (34), it can beshown that

$\begin{matrix}{b_{7} = {a_{7} - {2\left( {{\sqrt{5}a_{11}} + {\sqrt{7}a_{23}}} \right)\Delta\; u} - {2\left( {{\sqrt{10}a_{12}} - {\sqrt{5}a_{13}} + {\sqrt{14}a_{24}} - {\sqrt{7}a_{25}}} \right)\Delta\; v} + {5\sqrt{6}\left( {a_{16} + a_{17}} \right)\left( {\Delta\; u} \right)^{2}} + {10\sqrt{6}\left( {a_{18} - a_{19}} \right)\Delta\; u\;\Delta\; v} - {5\sqrt{6}\left( {a_{16} - {3a_{17}}} \right)\left( {\Delta\; v} \right)^{2}} - {10\sqrt{7}\left( {a_{22} + {2a_{23}}} \right)\left( {\Delta\; u} \right)^{3}} - {30\sqrt{7}\left( {{\sqrt{2}a_{24}} - a_{26}} \right)\left( {\Delta\; u} \right)^{2}\Delta\; v} + {30\sqrt{7}\left( {a_{22} - {2a_{23}}} \right)\Delta\;{u\left( {\Delta\; v} \right)}^{2}} - {10\sqrt{7}\left( {{3\sqrt{2}a_{24}} - {4a_{25}} + {a_{26}\left( {\Delta\; v} \right)}^{3}} \right.}}} & \left( {38a} \right) \\{b_{8} = {a_{8} - {2\left( {{\sqrt{10}a_{12}} + {\sqrt{5}a_{13}} + {\sqrt{14}a_{24}} + {\sqrt{7}a_{25}}} \right)\Delta\; u} - {2\left( {{\sqrt{5}a_{11}} + {\sqrt{7}a_{23}}} \right)\Delta\; v} + {5\sqrt{6}\left( {{3a_{18}} + a_{19}} \right)\left( {\Delta\; u} \right)^{2}} + {10\sqrt{6}\left( {a_{16} + a_{17}} \right)\Delta\; u\;\Delta\; v} + {5\sqrt{6}\left( {a_{18} - a_{19}} \right)\left( {\Delta\; v} \right)^{2}} - {10\sqrt{7}\left( {{3\sqrt{2}a_{24}} + {4a_{25}} + a_{26}} \right)\left( {\Delta\; u} \right)^{3}} - {30\sqrt{7}\left( {a_{22} + {2a_{23}}} \right)\left( {\Delta\; u} \right)^{2}\Delta\; v} - {30\sqrt{7}\left( {{\sqrt{2}a_{24}} - a_{26}} \right)\Delta\;{u\left( {\Delta\; v} \right)}^{2}} + {10\sqrt{7}\left( {a_{22} - a_{23}} \right)\left( {\Delta\; v} \right)^{3}}}} & \left( {38b} \right)\end{matrix}$

Equation (38) indicates that the primary (Z₁₂)and secondary (Z₂₄)spherical aberrations, the secondary (Z₁₁ and Z₁₃) and tertiary (Z₂₃ andZ₂₅) astigmatism, the secondary coma (Z₁₇ and Z₁₈), as well as Z₁₆, Z₁₉,Z₂₂, and Z₂₆ induce the coma when an ocular wavefront is decentered.

A primary spherical aberration (Z₁₂) Zernike polynomial typically doesnot induce trefoil. Other polynomials, such as those from Z₁₀ to Z₂₇,can contribute to the induction of trefoil, when an ocular wavefront isdecentered.

4.4 Wavefront Refraction of Decentered Aberrations

Because Zernike coefficients can change when the wavefront isdecentered, one thing to be noted is the change of the wavefrontrefraction. Indeed, there may be some discrepancy between the wavefrontrefraction and the manifest refraction, especially when the high orderaberrations are relatively significant. Because the ocular wavefront ismeasured in scotopic lighting condition and the manifest refraction ismeasured in mesopic to photopic lighting condition, not only may thepupil size change, but the pupil center may also shift.

With the use of Eq. (34), the second order Zernike coefficients can becalculated. Table 15 lists the second order Zernike coefficients ascontributed from the high order aberrations up to the sixth order. Ascan be seen each Zernike high order coefficient contributes to thesecond order Zernike coefficients when a decentration in both x- andy-direction occurs. In general, higher order coefficients have lesssignificant effect to the second order coefficients as they correspondto higher power of the decentration. Once the second order coefficientsare known, the wavefront refraction can be calculated by

$\begin{matrix}{{S = {{- \frac{4\sqrt{3}b_{2}^{0}}{R^{2}}} - \frac{2\sqrt{6}\sqrt{\left( b_{2}^{- 2} \right)^{2} + \left( b_{2}^{2} \right)^{2}}}{R^{2}}}},} & \left( {39a} \right) \\{{C = \frac{4\sqrt{6}\sqrt{\left( b_{2}^{- 2} \right)^{2} + \left( b_{2}^{2} \right)^{2}}}{R^{2}}},} & \left( {39b} \right) \\{\theta = {\frac{1}{2}{{\tan^{- 1}\left( \frac{b_{2}^{- 2}}{b_{2}^{2}} \right)}.}}} & \left( {39c} \right)\end{matrix}$

Equations (39a), (39b), and (39c) can be used as a basis for determininga refraction when decentration occurs. b values, such as b₂₀, can besubstituted from Table 15. To obtain an effect from decentration, theformulas from Table 15 can be used, which may depend on the originalZernike coefficient. The refraction could be influenced by almost allterms. Optionally, if a matrix formula such as Eq. (34) is used, bvalues can be obtained. For each case, such as a pupil constriction, arotation, a decentration, or any combination thereof, it is possible todetermine a general formula for the calculation of the new Zernikecoefficients. From the low order Zernike coefficients, i.e., c3, c4, andc5, the new wavefront refraction can be determined. Another set can beused to determine refractions, which are useful when dealing withaberrations.

According to some embodiments of the present invention, for example, itis possible to calculate the wavefront refraction for a 0.5 μm ofhorizontal coma (Z₈) and 0.5 μm of spherical aberration (Z₁₂) over a 6mm pupil when the pupil moves in the x-direction by 0.1 mm and 0.5 mm,respectively. From Table 15, we have b₃=0, b₄=−2√{square root over(6)}a₈Δu, and b₅=−2√{square root over (3)}a₈Δu for the case of coma.Substituting Δu=0.1/3=0.033 and Δu=0.5/3=0.165, respectively, into theseformulas, we have b₃=0, b₄=−2√{square root over (6)}×0.5×0.033=−0.081μm, and b₅=−2√{square root over (3)}×0.5×0.033=−0.057 μm. Using Eq.(39), we find the refraction as 0.031 DS/0.062 DC×0°. For a 0.5 mmdecentration, we obtain b₃=0, b₄=−0.404 μm, and b₅=−0.286 pm,corresponding to a refraction of 0.155 DS/0.311 DC×0°, which is exactly5 times the previous refraction when the decentration is 0.1 mm.

For the spherical aberration, we have b₃=0, b₄=4√{square root over(15)}a₁₂(Δu)², and b₅=2√{square root over (30)}a₁₂(Δu)². SubstitutingΔu=0.033 into these formulas, we get b₃=0, b₄=0.008 μm, b₅=0.006 μm.Using Eq. (39), we obtain the refraction as −0.009 DS/0.006 DC×0°. ForΔu=0.165, we have b₃=0, b₄=0.211 μm, and b₅=0.149 μm, corresponding to arefraction of −0.244 DS/0.162 DC×0°. Note that in the case of thespherical aberration, the refraction is no longer 5 times the previousrefraction when the decentration is 0.1 mm.

Table 15 lists Zernike coefficients of the defocus and astigmatism ascontributed from higher order Zernike coefficients when a wavefrontdecentration occurs. Note that the sphere and cylinder may not changewhen a decentration occurs when no high order aberrations exist. Forhigher order coefficients, the contribution can be a function of thedecentration in higher power: the powers of the decentration for the3rd, 4th, 5th, and 6th order coefficients are 1, 2, 3 and 4,respectively.

TABLE 15 b₃ = a₃ b₄ = a₄ b₅ = a₅ b₃ = −2{square root over (3)}(a₆ +a₇)Δu − 2{square root over (3)}(a₈ − a₉)Δv b₄ = −2{square root over(6)}(a₈Δu + a₇Δv) b₅ = −2{square root over (3)}(a₈ + a₉)Δu − 2{squareroot over (3)}(a₆ − a₇)Δv b₃ = 2{square root over (15)}(a₁₀ +2a₁₁)(Δu)² + 4{square root over (15)}({square root over (2)}a₁₂ −a₁₄)ΔuΔv − 2{square root over (15)}(a₁₀ − 2a₁₁)(Δv)² b₄ = 2{square rootover (15)}(2a₁₂ + {square root over (2)}a₁₃)(Δu)² + 4{square root over(30)}a₁₁ΔuΔv + 2{square root over (15)}(2a₁₂ − {square root over(2)}a₁₃)(Δv)² b₅ = 2{square root over (15)}(a₁₄ + {square root over(2)}a₁₂ + 2a₁₃)(Δu)² + 4{square root over (15)}a₁₀ΔuΔv − 2{square rootover (15)}(a₁₄ + {square root over (2)}a₁₂ − 2a₁₃)(Δv)² b₃ = 3{squareroot over (2)}(a₁₆ + a₁₇)Δu − 3{square root over (2)}(a₁₈ − a₁₉)Δv −10{square root over (2)}(a₁₅ + 3a₁₆ + 2a₁₇)(Δu)³ − 30{square root over(2)}(2a₁₈ − a₁₉ − a₂₀)(Δu)²Δv + 30{square root over (2)}(a₁₅ − a₁₆ +2a₁₇)Δu(Δv)² − 10{square root over (2)}(2a₁₈ − 3a₁₉ + a₂₀)(Δv)³ b₄ =−6a₁₈Δu − 6a₁₇Δv − 20(3a₁₈ + a₁₉)(Δu)³ − 60(a₁₆ + a₁₇)(Δu)²Δv − 60(a₁₈ −a₁₉)Δu(Δv)² + 20(a₁₆ − 3a₁₇)(Δv)³ b₅ = −3{square root over (2)}(a₁₈ +a₁₉)Δu − 3{square root over (2)}(a₁₆ − a₁₇)Δv − 10{square root over(2)}(4a₁₈ − 3a₁₉ + a₂₀)(Δu)³ − 30{square root over (2)}(a₁₅ +a₁₆)(Δu)²Δv − 30{square root over (2)}(a₁₉ − a₂₀)Δu(Δv)² + 10{squareroot over (2)}(a₁₅ − 3a₁₆ + 4a₁₇)(Δv)³ b₄ = 5{square root over(21)}(2a₂₄ + {square root over (2)}a₂₅(Δu)² + 10{square root over(42)}a₂₃ΔuΔv + 5{square root over (21)}(2a₂₄ − {square root over(2)}a₂₅)(Δv)² + 5{square root over (21)}(6a₂₄ + 4{square root over(2)}a₂₅ + {square root over (2)}a₂₆)(Δu)⁴ + 20{square root over(42)}(a₂₂ + 2a₂₃)(Δu)³Δv + 30{square root over (21)}(2a₂₄ − {square rootover (2)}a₂₆)(Δu)²(Δv)² − 20{square root over (42)}(a₂₂ − 2a₂₃)Δu(Δv)³ +5{square root over (21)}(6a₂₄ − 4{square root over (2)}a₂₅ + {squareroot over (2)}a₂₆)(Δv)⁴ b₅ = 5{square root over (21)}({square root over(2)}a₂₄ + 2a₂₅ + a₂₆)(Δu)² + 10{square root over (21)}a₂₂ΔuΔv − 5{squareroot over (21)}({square root over (2)}a₂₄ − 2a₂₅ + a₂₆)(Δv)² + 5{squareroot over (21)}(4{square root over (2)}a₂₄ + 7a₂₅ + 4a₂₆ + a₂₇)(Δu)⁴ +20{square root over (21)}(a₂₁ + 2a₂₂ + a₂₃)(Δu)³Δv + 30{square root over(21)}(a₂₅ − a₂₇)(Δu)²(Δv)² − 20{square root over (21)}(a₂₁ − 2a₂₂ +a₂₃)Δu(Δv)³ − 5{square root over (21)}(4{square root over (2)}a₂₄ −7a₂₅ + 4a₂₆ − a₂₇)(Δv)⁴

Wavefront RMS error and refractions can also be considered. If arefraction is close to zero, then there is a good opportunity forachieving a favorable result. A generic formula can indicate what thewavefront RMS error will be after correction. According to someembodiments, Eq. (39) provides such a generic formula. If there isdecentration that is not corrected for, then there is a greater chanceof having wavefront RMS error.

5. Wavefront Representation for Resizing, Rotation, and Decentration

Wavefront representation with Zernike polynomials has been discussed in,for example, Bará, S. et al., Appl. Opt., 39:3413-3420 (2000), Guirao,A. et al., J. Opt. Soc. Am. A, 18:1003-1015 (2001), Bará, S. et al., J.Opt. Soc. Am. A, 23:2061-2066 (2006), and Lundström L. et al., J. Opt.Soc. Am. A, 24:569-577 (2007). Lundström L. et al., J. Opt. Soc. Am. A,24:569-577 (2007) proposed the use of matrix transformations thatinclude pupil resizing, rotation, and decentration. However, thisapproach does not provide an analytical framework. Embodiments of thepresent invention, such as those exemplified in Tables 2, 10, 12, and15, provide an analytical framework that reveals physical insights onhow Zernike aberrations interact with each other when a geometricaltransformation takes place.

5.1 Wavefront Transformation with Zernike Polynomials

As discussed elsewhere herein, the conversion of Zernike coefficientscan be determined when a wavefront map goes through a geometricaltransformation, such as decentration, rotation, or pupil resizing. Whena combination of any of these happens, new Zernike coefficients can beobtained from the original set by the Zernike geometrical transformationmatrix asb=C^(egt) a,   (40)where the Zernike geometrical transformations matrix C^(egt) can bewritten as the multiplication of a series of conversion matrices asC^(egt)=C₃C₂C₁.   (41)

In Eq. (41), the matrices C₁, C₂, and C₃ represent the first, second,and the third geometrical transformations, respectively. They can be anyof the decentration matrix C^(z4z), the rotation matrix C^(z3z), or thepupil resizing matrix C^(z2z). The pupil resizing matrix C^(z2z) isrelated to Zernike resizing polynomials G_(n) ^(i)(ε).

As an example of wavefront decentration, rotation, and pupilconstriction, FIG. 21A shows an original wavefront 2100 a of 6 mm pupiland FIG. 21B shows the wavefront 2100 b when it undergoes a decentrationof −0.45 mm in the x- and 0.36 mm in the y-direction, respectively, arotation of 25° counter clockwise, and a pupil constriction to 4.8 mm.Put differently, FIG. 21B shows the wavefront after a decentration ofΔu=−0.15, Δv=−0.15, a rotation of 25° counter clockwise, and a pupilconstriction ratio of ε=0.8. The corresponding Zernike coefficientsafter each transformation are shown in Table 16.

Appendix E shows a Matlab code that implements Eq. (40) for any seriesof geometrical transformations. For the previous example, we haveΔu=−0.45/3=−0.15, Δv=0.36/3=0.12, φ=25π/180, and ε=4.8/6=0.8.Substituting these parameters into the function WavefrontTransform, thefinal Zernike coefficients can be obtained. The Zernike coefficientsafter each transformation can also be recorded, as shown in Table 16.Also shown in Table 16 are the total RMS wavefront error and high orderRMS wavefront error. It is interesting to note that after a rotation,both the total RMS error and the high order RMS error do not change. Inaddition, the spherical equivalent (S+C/2) also does not change.

5.2 Wavefront Refraction after Transformation

As shown in Table 16, any of the geometrical transformations may changethe low order Zernike coefficients, for example, b₃, b₄, and b₅ whenhigh order aberrations exist. Therefore, the wavefront refraction alsochanges. As discussed in the previous subsection, a new set of Zernikecoefficients can be calculated and Eq. (39) can be used to calculate thenew wavefront refraction.

For the same example as in the previous subsection, Table 16 shows thewavefront refraction in minus cylinder notation for the originalwavefront and after each of the geometrical transformations. In each ofthe geometrical transformations, the wavefront refraction changes.

Table 16 lists Zernike coefficients for the original wavefront, andthose after it decenters −0.45 mm in the x- and 0.36 mm in they-directions, respectively, and rotates by 25°, and finally its pupilconstricts to 4.8 mm, as shown in FIGS. 21A and 21B. The total RMS, highorder RMS, and the refractions in terms of sphere, cylinder, andcylinder axis are also shown. Minus cylinder notation is used.

TABLE 16 i n m Original Decentered Rotated Constricted  0 0 0 0.45011.2923 1.2923 1.0648  1 1 −1 −0.2689 −0.6344 0.4544 0.4739  2 1 1 2.07122.4358 2.4757 1.8950  3 2 −2 −0.8093 −0.8785 −0.6004 −0.5645  4 2 00.2609 0.3486 0.3486 0.2333  5 2 2 0.1310 −0.0466 0.6430 0.6530  6 3 −3−0.0218 0.0469 −0.2146 0.0541  7 3 −1 −0.2407 −0.1734 −0.1336 −0.0541  83 1 0.1607 0.0557 0.1238 0.2016  9 3 3 −0.0221 −0.2347 −0.1060 0.0786 104 −4 0.0462 0.1323 −0.1920 0.0089 11 4 −2 0.1168 0.3239 0.1873 0.2120 124 0 0.1687 −0.0212 −0.0212 0.0814 13 4 2 0.0953 −0.0273 −0.2657 −0.154814 4 4 −0.1079 −0.1717 −0.1005 0.0227 15 5 −5 −0.0314 0.1019 −0.1261−0.0413 16 5 −3 0.1452 0.0952 −0.1816 −0.0595 17 5 −1 0.1390 0.0504−0.0158 −0.0052 18 5 1 −0.0299 −0.1454 −0.1531 −0.0502 19 5 3 0.1312−0.2135 −0.1472 −0.0482 20 5 5 −0.1263 −0.0826 −0.0361 −0.0118 21 6 −6−0.0420 −0.0420 0.0493 0.0129 22 6 −4 0.0895 0.0895 −0.1003 −0.0263 23 6−2 −0.1400 −0.1400 −0.1550 −0.0406 24 6 0 −0.1032 −0.1032 −0.1032−0.0271 25 6 2 −0.0849 −0.0849 0.0527 0.0138 26 6 4 −0.0861 −0.0861−0.0732 −0.0192 27 6 6 0.0259 0.0259 −0.0014 −0.0004 RMS 2.3633 3.04883.0488 2.4273 HORMS 0.5296 0.6280 0.6280 0.3780 Sphere 0.30 0.33 0.210.25 Cylinder −0.71 −0.84 −0.60 −0.58 Axis 134° 135° 111° 108°

6. EXAMPLE 1

One example, according to embodiments of the present invention, involvesan eye that has −2.5 DS/+1.5 DC×81° and 0.35 microns of horizontal coma,−0.2 microns of vertical coma, and −0.28 microns of sphericalaberration, on a 6 mm pupil. It possible to determine how the refractionchanges when the wavefront decenters 0.2 mm in the x direction and 0.1mm in the y direction, when it is rotated by 30 degree counterclockwise, and when it is constricted to 5 mm pupil. The wavefront maps2200 a, 2200 b, 2200 c, and 2200 d are shown in FIGS. 22A to 22D,respectively and the refractions are shown afterwards. The wavefront mapof the original map is shown in FIG. 22A, and experiences a decentrationof 0.2 mm in the x and 0.1 mm in the y direction as shown in FIG. 22B, acyclorotation of 30 degree as shown in FIG. 22C, and a pupilconstriction from 6 mm to 5 mm as shown in FIG. 22D.

The following are the refractions:

Current −2.50 DS/+1.50 DC × 81° Decentered −2.48 DS/+1.62 DC × 81.8°Rotated −2.48 DS/+1.62 DC × 111.8° Constricted −2.74 DS/+1.62 DC ×111.8°

Without being bound by any particular theory, it is thought that apossible reason that the cylinder does not change is because there areonly high order terms that affects the sphere. If the secondaryastigmatism were present, for example, the cylinder would have changed.

7. EXAMPLE 2

In another example, according to embodiments of the present invention,it can be shown that in these geometrical transformations, which termscontributed the most to the sphere power and which terms to the cylinderpower.

7.1 Decentration

The influence of higher order aberrations on the refraction due towavefront decentration may in some embodiments be a bit uncertain orcomplicated. However, in general, because the decentration is often muchsmaller than the pupil radius, the influence is dominated by terms thathave the lowest powers of the decentration. For example, for defocus,the influence comes from the coma, primary spherical aberration, andsecondary astigmatism, among other high order aberrations. Coma has themost impact on the refraction because it is linearly related to thedecentration. But for spherical aberration and secondary astigmatism,the relation to the decentration is quadratic, although the coefficientfor spherical aberration is about two times larger.

7.2 Rotation

When a wavefront map rotates, the sphere and cylinder typically do notchange. Generally, only the cylinder axis changes by an additional angleof the rotation.

7.3 Pupil Constriction

The defocus, or the spherical equivalent, can be affected by theprimary, secondary, tertiary, and higher order spherical aberrations.The cylinder can be affected by the primary, secondary, tertiary, andhigher order astigmatisms. For example, influence of the primary,secondary, and tertiary spherical aberration (SA) on sphericalequivalent, or defocus, is shown in FIG. 23. Apparently, when the pupilconstricts over half (pupil<3 mm), the influence from the primary,secondary, and tertiary SA becomes larger when it goes to higher orders.FIG. 23 shows that when pupil constricts, a positive sphericalaberration will make the refraction more and more hyperopicmonotonically. On the other hand, a negative spherical aberration maymake it more myopic as the pupil constricts. For the secondary SA,however, for a positive spherical aberration, the refraction can becomeslightly more hyperopic initially as the pupil constricts, but canquickly become more myopic as the constriction continues. For a negativesecondary SA, the situation can exactly reverse. For a positive tertiarySA, the refraction initially can become more myopic, then more hyperopicbefore it can become more hyperopic again finally. Again, for thenegative tertiary SA, the situation can be reversed. This is why forpresbyopic correction, in some embodiments it may be desirable tointroduce negative primary SA, positive secondary SA, and negativetertiary SA so when the pupil constricts, it becomes more myopic.

In a situation when the wavefront maps changes due to decentration,pupil constriction, or rotation, during the surgery, it is possible torecalculate a new map and deliver the ablation based on that new map.

Induced High Order Aberrations Corresponding to GeometricalTransformations, and Techniques for Determining, Reporting, and Using inTreatment

Embodiments of the present invention encompass systems, methods, andcomputer program products for evaluating induced high order aberrationscorresponding to geometrical transformations such as cyclorotation,pupil center shift, pupil constriction, and pupil dilation. Embodimentsalso encompass techniques for treating and reporting ocular aberrationsover scaled or decentered pupils, or other geometrical transformationsassociated with pupils. In some instances, the techniques can beimplemented in connection with software residing in or associated withsystems such as WaveScan®, iDesign™, or CustomVue® devices.

FIG. 24 provides a flow chart illustrating aspects of the use ofdiagnostic display and the use of retreatment and enhancement treatmentby means of a pupil rescaling algorithm. As shown here, system 2400 caninclude an input module 2405 that receives Zernike coefficientinformation, an input module 2410 that receives wavefront diameterinformation, a manifest refraction environment module 2415 that receivesmanifest refraction environment information, and a user choice module2420 that receives user choice information. Zernike coefficientinformation can include information from a wavefront examination, andencompass low order and high order aberrations. Manifest refractionenvironment information can encompass data related to lightingconditions. User choice information can encompass data related tophysicians or operators, for example according to their needs orpreferences.

A desired pupil size module 2425 can determine a desired pupil sizebased on manifest refraction environment information and user choiceinformation received from manifest refraction environment module 2415and user choice module 2420, respectively. A Zernike scaling algorithmmodule 2430 can operate to determine scaled Zernike coefficients basedon Zernike coefficient information received from input module 2405,wavefront diameter information received from input module 2410, anddesired pupil size information received from desired pupil size module2425. In some cases, the desired pupil size can be 4 mm, 5 mm, 6 mm, or7 mm. In some cases, the desired pupil size can be a scotopic pupilsize, a mesopic pupil size, or a photopic pupil size. Optionally, apupil size can be equivalent to the pupil size when the manifestrefraction is taken.

As shown here, Zernike scaling algorithm module 2430 can output a set ofscaled Zernike coefficients 2435. High order aberration module 2440 canoperate to determine high order aberrations based on the scaled Zernikecoefficient information 2435. Similarly, wavefront refraction module2445 can operate to determine a wavefront refraction based on the scaledZernike coefficient information 2435. A predicted night vision symptommodule 2450 can determine predicted night vision symptoms based on highorder aberration information received from high order aberration module2440. A potential treatment or retreatment module 2455 can determine apotential treatment or retreatment based on wavefront refractioninformation received from wavefront refraction module 2445 and predictednight vision symptom information received from predicted night visionsymptom module 2450. In some cases, a predicted night vision symptom caninclude a spherical aberration. Treatment module 2455 can be configuredto reduce low order aberrations (e.g. as determined from a wavefrontrefraction), and to reduce night vision symptoms.

FIG. 25 provides a flow chart showing aspects of diagnostic displaytechniques for comparing or differencing among different exams from thesame eye at different timestamps (which may be on the same day, or ondifferent days) or for different eyes. As shown here, system 2500 caninclude a first module 2505 that receives Zernike coefficientinformation related to a first examination, and a second or n^(th)module 2510 that receives Zernike coefficient information related to asecond or n^(th) examination. System 2500 also includes a minimumwavefront diameter module 2515 that operates to determine a minimumwavefront diameter based on or corresponding to information receivedfrom the first module 2505, the second module 2510, or both. System 2500further includes a pupil scaling algorithm module 2520 that can generatescaled information based on the minimum wavefront diameter informationreceived from minimum wavefront diameter module 2515. FIG. 25 alsoillustrates an analysis module 2530, which can operate to compare,difference, or display aspects of the various examinations, optionallybased scaled information received from pupil scaling algorithm module2520. As shown here, analysis module 2530 includes a first scaledZernike coefficient module 2535 that generates or processes scaledZernike coefficient information related to the first examination, and asecond or n^(th) scaled Zernike coefficient module 2540 that generatesor processes scaled Zernike coefficient information related to thesecond or n^(th) examination.

The systems and procedures described in FIGS. 24 and 25 can incorporatescaling or other features described in previously incorporated U.S.patent application Ser. No. 12/722,881, filed Mar. 12, 2010.

FIG. 26 provides a flow chart illustrating aspects of the use ofdiagnostic display and the use of retreatment and enhancement treatmentby means of a geometrical transformation algorithm. For example, adifference between a first geometrical configuration and a secondgeometrical configuration can include a cyclorotation, a pupil centershift or decentration (e.g. x- and y-shifts), a pupil constriction, apupil dilation, or any combination thereof. As shown here, system 2600can include an input module 2605 that receives Zernike coefficientinformation, an input module 2610 that receives wavefront diameterinformation, a manifest refraction environment module 2615 that receivesmanifest refraction environment information, and a user choice module2620 that receives user choice information. Zernike coefficientinformation can include information from a wavefront examination, andencompass low order and high order aberrations. Manifest refractionenvironment information can encompass data related to lightingconditions. User choice information can encompass data related tophysicians or operators, for example according to their needs orpreferences.

A desired x- and y-shift, rotation, or pupil size module 2625 candetermine a desired x- and y-shift, rotation, or pupil size change basedon manifest refraction environment information and user choiceinformation received from manifest refraction environment module 2615and user choice module 2620, respectively. A geometrical transformationalgorithm module 2630 can operate to determine transformed Zernikecoefficients based on Zernike coefficient information received frominput module 2605, wavefront diameter information received from inputmodule 2610, and desired x- and y-shift, rotation, or pupil size changeinformation received from desired x- and y-shift, rotation, or pupilsize change module 2625.

As shown here, geometrical transformation algorithm module 2630 canoutput a set of transformed Zernike coefficients 2635. High orderaberration module 2640 can operate to determine high order aberrationsbased on the transformed Zernike coefficient information 2635.Similarly, wavefront refraction module 2645 can operate to determine awavefront refraction based on the transformed Zernike coefficientinformation 2635. According to some embodiments, a high order aberrationcan be induced by various geometrical transformations, includingcyclorotation, pupil center shift or decentration, pupil constriction,or pupil dilation. A predicted night vision symptom module 2650 candetermine predicted night vision symptoms based on high order aberrationinformation received from high order aberration module 2640. A potentialtreatment or retreatment module 2655 can determine a potential treatmentor retreatment based on wavefront refraction information received fromwavefront refraction module 2645 and predicted night vision symptominformation received from predicted night vision symptom module 2650. Insome cases, a predicted night vision symptom can include a sphericalaberration. Treatment module 2655 can be configured to reduce low orderaberrations (e.g. as determined from a wavefront refraction), and toreduce night vision symptoms.

FIG. 27 provides a flow chart showing aspects of diagnostic displaytechniques for comparing or differencing among different exams from thesame eye at different timestamps (which may be on the same day, or ondifferent days) or for different eyes. As shown here, system 2700 caninclude a first module 2705 that receives Zernike coefficientinformation related to a first examination (e.g. at Time 1), and asecond or n^(th) module 2710 that receives Zernike coefficientinformation related to a second or n^(th) examination (e.g. Time 2 orn). System 2700 also includes a minimum wavefront diameter module 2715that operates to determine a minimum wavefront diameter based on orcorresponding to information received from the first module 2705, thesecond module 2710, or both. System 2700 further includes a geometricaltransformation algorithm module 2720 that can generate scaledinformation based on the minimum wavefront diameter information receivedfrom minimum wavefront diameter module 2715. FIG. 27 also illustrates ananalysis module 2730, which can operate to compare, difference, ordisplay aspects of the various examinations, optionally based scaledinformation received from geometrical transformation algorithm module2720. As shown here, analysis module 2730 includes a first scaledZernike coefficient module 2735 that generates or processes scaledZernike coefficient information related to the first examination, and asecond or n^(th) scaled Zernike coefficient module 2740 that generatesor processes scaled Zernike coefficient information related to thesecond or n^(th) examination.

The systems and procedures described in FIGS. 26 and 27 can incorporategeometrical transformation or other features described in previouslyincorporated U.S. patent application Ser. No. 12/725,575, filed Mar. 17,2010.

FIG. 28 illustrates aspects of a method 2800 of determining a high orderaberration induced by a change in geometrical configuration in an eye ofa patient. As shown here, the method includes the steps of inputting afirst geometrical configuration of the eye 2810, and inputting anoriginal set of coefficients for a basis function characterizing thefirst geometrical configuration of the eye 2820. The basis function canbe separated into a product of a first set of radial polynomials and afirst triangular function. Method 2800 may also include the steps ofinputting a second geometrical configuration of the eye 2830, andinputting a transformed set of coefficients for the basis function 2840.The transformed set of coefficients can be based on the firstgeometrical configuration of the eye, the original set of coefficients,and the second geometrical configuration of the eye. Method 2800 canalso include the step of determining an induced high order aberrationbased on the transformed set of coefficients 2850.

Each of the above calculations or operations may be performed using acomputer or other processor having hardware, software, and/or firmware.The various method steps may be performed by modules, and the modulesmay comprise any of a wide variety of digital and/or analog dataprocessing hardware and/or software arranged to perform the method stepsdescribed herein. The modules optionally comprising data processinghardware adapted to perform one or more of these steps by havingappropriate machine programming code associated therewith, the modulesfor two or more steps (or portions of two or more steps) beingintegrated into a single processor board or separated into differentprocessor boards in any of a wide variety of integrated and/ordistributed processing architectures. These methods and systems willoften employ a tangible media embodying machine-readable code withinstructions for performing the method steps described above. Suitabletangible media may comprise a memory (including a volatile memory and/ora non-volatile memory), a storage media (such as a magnetic recording ona floppy disk, a hard disk, a tape, or the like; on an optical memorysuch as a CD, a CD-R/W, a CD-ROM, a DVD, or the like; or any otherdigital or analog storage media), or the like.

All patents, patent publications, patent applications, journal articles,books, technical references, and the like discussed in the instantdisclosure are incorporated herein by reference in their entirety forall purposes.

While the exemplary embodiments have been described in some detail, byway of example and for clarity of understanding, those of skill in theart will recognize that a variety of modification, adaptations, andchanges may be employed. Hence, the scope of the present inventionshould be limited solely by the claims.

Appendix A Zernike Resizing Polynomials

Zernike resizing polynomials are the basis for calculating the resizedZernike coefficients from an original set of Zernike coefficients whenthe pupil size is changed. Following are some properties of this set ofpolynomials.

Property 1.

${G_{n}^{i}(ɛ)} = {{\frac{1}{\sqrt{\left( {n + 1} \right)}}\left\lbrack {{\mathcal{R}_{n + {2i}}^{n}(ɛ)} - {\mathcal{R}_{n + {2i}}^{n + 2}(ɛ)}} \right\rbrack}.}$

-   Proof: This relationship can be proved by dividing √{square root    over (n+1)} on both sides of Eq. (A1) and comparing the result to    Zernike resizing polynomials Eq. (A2).

$\begin{matrix}{{{\mathcal{R}_{n + {2i}}^{n}(ɛ)} - {\mathcal{R}_{n + {2i}}^{n + 2}(ɛ)}} = {{ɛ^{n}\left( {n + 1} \right)}\sqrt{n + {2i} + 1}{\sum\limits_{j = 0}^{i}{\frac{\left( {- 1} \right)^{i + j}{\left( {n + i + j} \right)!}}{{j!}{\left( {i - j} \right)!}{\left( {n + j + 1} \right)!}}{ɛ^{2j}.}}}}} & ({A1}) \\{\mspace{79mu}{{G_{n}^{i}(ɛ)} = {ɛ^{n}\sqrt{\left( {n + {2i} + 1} \right)\left( {n + 1} \right)}{\sum\limits_{j = 0}^{i}{\frac{\left( {- 1} \right)^{i + j}{\left( {n + i + j} \right)!}}{{\left( {n + j + 1} \right)!}{\left( {i - j} \right)!}{j!}}{ɛ^{2j}.}}}}}} & ({A2})\end{matrix}$Property 2. G_(n) ^(i)(1)=0 for i≠0.

-   Proof From Property 1, we have

$\begin{matrix}{{{G_{n}^{i}(1)} = {{\frac{1}{\sqrt{\left( {n + 1} \right)}}\left\lbrack {{\mathcal{R}_{n + {2\; i}}^{n}(1)} - {\mathcal{R}_{n + {2i}}^{n + 2}(1)}} \right\rbrack} = {{\frac{1}{\sqrt{n + 1}}\left( {\sqrt{n + 1} - \sqrt{n + 1}} \right)} = 0}}},} & ({A3})\end{matrix}$because for any n and i except i=0 it can be shown [Born, M. and Wolf,E., Principles of Optics, 5th ed. (Cambridge University Press, 1999),Chap 9]

_(n) ^(|m|)(1)=√{square root over (n+1)}.   (A4)Property 3. G_(n) ⁰(ε)=1.

-   Proof: Because for i=0,    _(n) ^(n+2)(ε)=0, from Property 1 we have

$\begin{matrix}{{G_{n}^{0}(ɛ)} = {{\frac{1}{\sqrt{n + 1}}{\mathcal{R}_{n}^{n}(ɛ)}} = {\frac{ɛ^{n}\sqrt{n + 1}}{\sqrt{n + 1}} = {ɛ^{n}.}}}} & ({A5})\end{matrix}$Appendix B Derivation of Eq. (27)

The wavefront after cyclorotation of angle φ, as shown in FIG. 10,represented by Taylor monomials in Cartesian coordinates, can be givenas

$\begin{matrix}{{W\left( {\rho,{\theta;\phi}} \right)} = {\sum\limits_{p,q}{a_{p}^{q}\rho^{p}{\cos^{q}\left( {\theta - \phi} \right)}{{\sin^{p - q}\left( {\theta - \phi} \right)}.}}}} & ({B1})\end{matrix}$

The Taylor monomials in the original coordinates can be written as

$\begin{matrix}\begin{matrix}{{T_{p}^{q}\left( {\rho,{\theta;\phi}} \right)} = {T_{p}^{q}\left( {\rho,{\theta - \phi}} \right)}} \\{= {{\rho^{p}\left\lbrack {\cos\left( {\theta - \phi} \right)} \right\rbrack}^{q}\left\lbrack {\sin\left( {\theta - \phi} \right)} \right\rbrack}^{p - q}} \\{= {{\rho^{p}\left\lbrack {{\cos\;\theta\;\cos\;\phi} + {\sin\;\theta\;\sin\;\phi}} \right\rbrack}^{q}\left\lbrack {{\sin\;\theta\;\cos\;\phi} - {\cos\;\theta\;\sin\;\phi}} \right\rbrack}^{p - q}} \\{= {\sum\limits_{k = 0}^{q}{\sum\limits_{l - 0}^{p - q}{\frac{\left( {- 1} \right)^{l}{q!}{\left( {p - q} \right)!}}{{k!}{l!}{\left( {q - k} \right)!}{\left( {p - q - l} \right)!}}\left( {\cos\;\theta} \right)^{q - k + l}\left( {\sin\;\theta} \right)^{p - q + k - l} \times}}}} \\{\left( {\sin\;\phi} \right)^{k + l}\left( {\cos\;\phi} \right)^{p - k - l}} \\{= {\sum\limits_{k = 0}^{q}{\sum\limits_{l - 0}^{p - q}{\frac{\left( {- 1} \right)^{l}{q!}{\left( {p - q} \right)!}}{{k!}{l!}{\left( {q - k} \right)!}{\left( {p - q - l} \right)!}} \times}}}} \\{\left( {\sin\;\phi} \right)^{k + l}\left( {\cos\;\phi} \right)^{p - k - l}{{T_{p}^{q - k + 1}\left( {\rho,\theta} \right)}.}}\end{matrix} & ({B2})\end{matrix}$Therefore, the rotated Taylor coefficients b_(p) ^(q) is related to theoriginal Taylor coefficients a_(p) ^(q) by changing φ to −φ in Eq. (B2)as

$\begin{matrix}{b_{p}^{q} = {\sum\limits_{k = 0}^{q}{\sum\limits_{l - 0}^{p - q}{\frac{\left( {- 1} \right)^{k}{q!}{\left( {p - q} \right)!}}{{k!}{l!}{\left( {q - k} \right)!}{\left( {p - q - l} \right)!}}\left( {\sin\;\phi} \right)^{k + l}\left( {\cos\;\phi} \right)^{p - k - l}{a_{p}^{q - k + l}.}}}}} & ({B3})\end{matrix}$Appendix C Derivation of Eq. (28)

To derive Eq. (28), it is helpful to start with the definition ofZernike polynomialsZ _(n) ^(m)(ρ,θ)=

_(n) ^(|m|)(ρ)Θ^(m)(θ),   (C1)where the triangular function

$\begin{matrix}{{\Theta^{m}(\theta)} = \left\{ \begin{matrix}{\sin{m}\theta} & \left( {m < 0} \right) \\1 & \left( {m = 0} \right) \\{\cos{m}\theta} & {\left( {m > 0} \right).}\end{matrix} \right.} & ({C2})\end{matrix}$

Consider a pair of terms, i.e., with the same radial order n butopposite sign of azimuthal frequency m. The Zernike terms of the rotatedwavefront can be written as

$\begin{matrix}{{\mathcal{R}_{n}^{m}\left\lbrack {{a_{n}^{- {m}}\sin{m}\left( {\theta - \phi} \right)} + {a_{n}^{m}\cos{m}\left( {\theta - \phi} \right)}} \right\rbrack} = {{\mathcal{R}_{n}^{m}\left\lbrack {{a_{n}^{- {m}}\left( {{\sin{m}\theta\;\cos{m}\phi} - {\cos{m}\theta\;\sin{m}\phi}} \right)} + {a_{n}^{m}\left( {{\cos{m}\theta\;\cos{m}\phi} + {\sin{m}\theta\;\sin{m}\phi}} \right)}} \right\rbrack} = {{\mathcal{R}_{n}^{m}\left\lbrack {{\left( {{a_{n}^{- {m}}\sin{m}\phi} + {a_{n}^{m}\cos{m}\phi}} \right)\sin{m}\theta} + {\left( {{a_{n}^{m}\cos{m}\phi} - {a_{n}^{- {m}}\sin{m}\phi}} \right)\cos{m}\theta}} \right\rbrack} = {{\mathcal{R}_{n}^{m}\left( {{b_{n}^{- {m}}\sin{m}\theta} + {b_{n}^{m}\cos{m}\theta}} \right)}.}}}} & ({C3})\end{matrix}$

From these last two lines of Eq. (C3), we haveb _(n) ^(−|m|) =a _(n) ^(−|m|)sin|m|φ+a _(n) ^(|m|cos|) m|φ,   (C4a)b _(n) ^(|m|) =a _(n) ^(|m|)cos|m|φ−a _(n) ^(−|m|)sin|m|φ,   (C4b)Appendix D Derivation of Eq. (32)Suppose an ocular wavefront is represented by a set of Taylorcoefficients {a_(p′) ^(q′)}. When it is decentered by Δu and Δv, we have

$\begin{matrix}\begin{matrix}{W = {\sum\limits_{i = 0}^{J}{a_{i}{T_{i}\left( {{u - {\Delta\; u}},{v - {\Delta\; v}}} \right)}}}} \\{= {\sum\limits_{i = 0}^{J}{{a_{i}\left( {u - {\Delta\; u}} \right)}^{q}\left( {v - {\Delta\; v}} \right)^{p - q}}}} \\{= {\sum\limits_{i = 0}^{J}{a_{i}{\sum\limits_{k = 0}^{q}{\sum\limits_{l = 0}^{p - q}{\frac{\left( {- 1} \right)^{k + 1}{q!}{\left( {p - q} \right)!}}{{k!}{l!}{\left( {q - k} \right)!}{\left( {p - q - l} \right)!}}\left( {\Delta\; u} \right)^{k}\left( {\Delta\; v} \right)^{l}{{T_{p - k - l}^{q - k}\left( {u,v} \right)}.}}}}}}}\end{matrix} & ({D1})\end{matrix}$

In order to obtain the new coefficients b_(p′) ^(q′), it is helpful tomake the following conversionp′=p−k−l,   (D2a)q′=q−k.   (D2b)

Solving Eq. (D2a, D2b) for k and l, we getk=q−q′,   (D3a)l=p−p′−(q−q′).   (D3b)Substituting k and l back to Eq. (D1), we obtain

$\begin{matrix}{b_{p}^{q}{\sum\limits_{p^{\prime},q^{\prime}}{\frac{\left( {- 1} \right)^{p - p^{\prime}}{q!}{\left( {p - q} \right)!}}{{\left( {q - q^{\prime}} \right)!}{\left( {p - p^{\prime} - q + q^{\prime}} \right)!}{\left( q^{\prime} \right)!}{\left( {p^{\prime} - q^{\prime}} \right)!}}\left( {\Delta\; u} \right)^{q - q^{\prime}}\left( {\Delta\; v} \right)^{p - p^{\prime} - q + q^{\prime}}{a_{p^{\prime}}^{q^{\prime}}.}}}} & ({D4})\end{matrix}$Appendix E Matlab Code for Geometrical Transformations

% This function calculate a new set of Zernike coefficients from anoriginal set when a % decentration of (du, dv), a rotation of phicounter clockwise, and a pupil resizing of % e occur. % function B =WavefrontTransform(A, du, dv, phi, e);     B = Z4Z(A, du, dv);     B =Z3Z(B, phi);     B = Z2Z(B, e); % This function converts an original setof Zernike coefficients to a new set when the pupil % size changesfunction B = Z2Z(A, e);   for i = 0:length(A)−1     [n, m] =single2doubleZ(i);     B(i+1) = getB(A, n, m, e);   end % This functioncalculates Zernike coefficients as the pupil resizes % function b =getB(A, n, m, e);   [N, M] = single2doubleZ(length(A)−1); x = 0;   for i= 1:(N−n)/2     y = 0;     for j = 0:i       z = 1;       for k = 0:i−2        z = z * (n+j+k+2);       end       y = y + (−1){circumflex over( )}(i+j)/factorial(i−j)/factorial(j)*z*e{circumflex over ( )}(2*j);    end     jj = double2singleZ(n+2*i, m);     x = x +sqrt((n+2*i+1)*(n+1))*y*A*(jj+1);   end   jj = double2singleZ(n, m);   b= (A(jj+1) + x)*e{circumflex over ( )}n; % This function converts Taylorcoefficients as map shifts by du, dv % function B = T4T(A, du, dv);  for i = 0:length(A)−1     B(i+1) = 0;     [p, q] = single2doubleT(i);    for j = 0:length(A)−1       [p2, q2] = single2doubleT(j);      if(p2 >= p && q2 >= q && p2−p−q2+q >= 0)        cc =(−1){circumflex over ( )}(p2−p)*factorial(q2)*factorial(p2−q2)/(...        factorial(q2−q)*factorial(p2−p−q2+q)*factorial(q) ...        *factorial(p−q));         B(i+1) = B(i+1) + cc*(du){circumflexover ( )}(q2−q)*(dv){circumflex over ( )}(p2−p−q2+q) ...        *A(j+1);       end     end   end % This function convertsZernike coefficients when map shifts du, dv % function B = Z4Z(A, du,dv);   A = Z2T(A);   B = T4T(A, du, dv);   B = T2Z(B); % This functioncalculates Zernike coefficients when map rotates phi % function B =Z3Z(A, phi);   for i = 1:length(A)−1     [n, m] = single2doubleZ(i);    jj1 = double2singleZ(n, −abs(m));     jj2 = double2singleZ(n,abs(m));     if(m < 0)      B(i+1) =A(jj1+1)*cos(m*phi)+A(jj2+1)*sin(−m*phi);     else      B(i+1) =A(jj1+1)*sin(m*phi)+A(jj2+1)*cos(m*phi);     end   end   B(1) = A(1); %This function converts Taylor coefficients to Zernike coefficients %function A = T2Z(B);   for i = 0:length(B)−1     [n, m] =single2doubleZ(i);     A(i+1) = 0;     for j = 0:length(B)−1       [p,q] = single2doubleT(j);       %% Now calculating the first summation      s1 = 0;       for ss = 0:(n−abs(m))/2         s1 = s1 +(−1){circumflex over ( )}ss*factorial(n−ss)/factorial(ss)/ ...        (n+p−2*ss+2)/factorial((n+m)/2−ss)/factorial( ...        (n−m)/2−ss);       end       s1 = s1*sqrt(n+1);       %% Nowcalculating the second summation       s2 = 0;             for t = 0:q              a = factorial(t);               b = factorial(q−t);              for t2 = 0:p−q                 c = factorial(t2);                d = factorial(p−q−t2);                 if (m >= 0 &&mod(p−q, 2) = = 0)                  s2 = s2 + 2*(−1){circumflex over( )}((p−q)/2+t2)/(a*b*c*d);                 elseif (p−2*t−2*t2 = =m||p−2*q−2*t2+2*t = = m)                   s2 = s2 + (−1){circumflexover ( )}((p−q)/2=t2)/(a*b*c*d);                 end              elseif (m < 0 && mod(p−q, 2) = = 1)                 if(p−2*q+2*t−2*t2= =−m && 2*q−p+2*t2−2*t = −m)                  s2 = s2 +(−1){circumflex over ( )}((p−q−1)/2+t2)/(a*b*c*d);                elseif (2*q−p−2*t+2*t2−m&&p−2*q−2*t2+2*t= =m)                 s2 = s2 − (−1){circumflex over( )}((p−q−1)/2+t2)/(a*b*c*d);                 end               end            end         end         if (m = = 0)          s2 =s2*factorial(q)*factorial(p−q)/2{circumflex over ( )}p;         else         s2 = sqrt(2)*s2*factorial(q)*factorial(p−q)/2{circumflex over( )}p;         end         A(i+1) = A(i+1) + B(j+1)*s1*s2;       end    end % This function converts Zernike coefficients to Taylorcoefficients % function B = Z2T(A);   B = zeros(1, length(A));   for i =0:length(A)−1     [n, m] = single2doubleZ(i);     for j = 0:length(A)−1      [p, q] = single2doubleT(j);       if (n < p || mod(n−p,2)= =1 ||mod(p−abs(m),2) = =1)         continue;       end       ss = 0;      facl = (−1){circumflex over( )}((n−p)/2)*sqrt(n+1)/factorial((n−p)/2) ...    /factorial((p+abs(m))/2)*factorial((n+p)/2) ...    factorial(abs(m));     tt2 = (p−abs(m))/2;     if (m > 0)       tt =floor(abs(m)/2);       norm = sqrt(2);     elseif (m == 0)       tt = 0;      norm = 1;     else       tt = floor((abs(m)−1)/2);       norm =sqrt(2);     end     sss = 0;     for t = 0:tt       for  t2 = 0:tt2        if (t+t2 = = (p−q)/2 && m >= 0)          ss = (−1){circumflexover ( )}t*norm/factorial(t2)/factorial ...         (2*t)/factorial((p−abs(m))/2−t2) ...         /factorial(abs(m)−2*t);          sss = sss + ss;       elseif(t+t2 = = (p−q−1)/2 && m < 0)          ss = (−1){circumflex over( )}t*norm/factorial(t2)/factorial ...         (2*t+1)/factorial((p−abs(m))/2−t2) ...         /factorial(abs(m)−2*t−1);          sss = sss + ss;         end      end     end     ss = sss*fac1;     j = double2singleT(p, q);    if (j >= 0)       B(j+1) = B(j+1) + ss*A(i+1);     end   end end %This function converts single → double index in Zernike polynomials %function [n, m] = single2doubleZ(jj);     n = floor(sqrt(2*jj+1)+0.5)−1;    m = 2*jj−n*(n+2); % This function converts double->single index inZernike polynomials % function jj = double2singleZ(n, m);     jj =(n{circumflex over ( )}2+2*n+m)/2; % This function converts single todouble index in Taylor monomials % function [p, q] = single2doubleT(jj);    p = floor((sqrt(1+8*jj)−1)/2);     q = jj−p*(p+1)/2; % This functionconverts double to single index in Taylor monomials % function jj =double2singleT(p, q);               jj = p*(p+1)/2+q;

What is claimed is:
 1. A method of determining a high order aberrationinduced by a change in geometrical configuration in an eye of a patient,comprising, through the use of a computer processor: inputting a firstgeometrical configuration of the eye; inputting an original set ofcoefficients for a basis function characterizing the first geometricalconfiguration of the eye, wherein the basis function can be separatedinto a product of a first set of radial polynomials and a firsttriangular function; inputting a second geometrical configuration of theeye; inputting a transformed set of coefficients for the basis function,wherein the transformed set of coefficients are based on the firstgeometrical configuration of the eye, the original set of coefficients,and the second geometrical configuration of the eye, such that eachcoefficient of the transformed set of coefficients is based on acorresponding coefficient of the original set of coefficients and acorresponding polynomial; and determining the induced high orderaberration based on the transformed set of coefficients.
 2. The methodaccording to claim 1, wherein a difference between the first geometricalconfiguration and the second geometrical configuration comprises a pupilcenter shift.
 3. The method according to claim 1, wherein a differencebetween the first geometrical configuration and the second geometricalconfiguration comprises a pupil constriction or a pupil dilation.
 4. Themethod according to claim 1, wherein a difference between the firstgeometrical configuration and the second geometrical configurationcomprises a cyclorotation and a member selected from the groupconsisting of a pupil center shift, a pupil constriction, and a pupildilation.
 5. The method according to claim 1, wherein the basis functioncomprises a Zemike basis function.
 6. The method according to claim 1,wherein the basis function comprises a Taylor basis function.
 7. Themethod according to claim 1, wherein the basis function comprises aSeidel basis function.
 8. The method according to claim 1, wherein theinduced high order aberration comprises a member selected from the groupconsisting of coma, secondary coma, trefoil, primary sphericalaberration, secondary spherical aberration, secondary astigmatism, andtertiary astigmatism.
 9. The method according to claim 1, furthercomprising determining a predicted vision symptom based on the inducedhigh order aberration.
 10. The method according to claim 9, wherein thevision symptom comprises a predicted night vision symptom.
 11. Themethod according to claim 1, further comprising determining a treatmentbased on the induced high order aberration.
 12. The method according toclaim 1, further comprising displaying the transformed set ofcoefficients for the basis function.
 13. The method according to claim1, further comprising displaying the induced high order aberration. 14.A system for determining a high order aberration induced by a change ingeometrical configuration in an eye of a patient, the system comprising:a first module that accepts a first geometrical configuration of theeye; a second module that accepts an original set of coefficients for abasis function characterizing the first geometrical configuration of theeye, wherein the basis function can be separated into a product of afirst set of radial polynomials and a first triangular function; a thirdmodule that accepts a second geometrical configuration of the eye; afourth module that determines a transformed set of coefficients for thebasis function, wherein the transformed set of coefficients are based onthe first geometrical configuration of the eye, the original set ofcoefficients, and the second geometrical configuration of the eye, suchthat each coefficient of the transformed set of coefficients is based ona corresponding coefficient of the original set of coefficients and acorresponding polynomial; and a fifth module that determines the inducedhigh order aberration based on the transformed set of coefficients. 15.The system according to claim 14, wherein a difference between the firstgeometrical configuration and the second geometrical configurationcomprises a pupil center shift.
 16. The method according to claim 14,wherein a difference between the first geometrical configuration and thesecond geometrical configuration comprises a pupil constriction or apupil dilation.
 17. The method according to claim 14, wherein adifference between the first geometrical configuration and the secondgeometrical configuration comprises a cyclorotation and a memberselected from the group consisting of a pupil center shift, a pupilconstriction, and a pupil dilation.
 18. A computer program product fordetermining a high order aberration induced by a change in geometricalconfiguration in an eye of a patient, the computer program productcomprising: code for accepting a first geometrical configuration of theeye; code for determining an original set of coefficients for a basisfunction characterizing the first geometrical configuration, wherein thebasis function can be separated into a product of a first set of radialpolynomials and a first triangular function; code for accepting a secondgeometrical configuration of the eye; code for determining a transformedset of coefficients for the basis function, wherein the transformed setof coefficients are based on the first geometrical configuration of theeye, the original set of coefficients, and the second geometricalconfiguration of the eye, such that each coefficient of the transformedset of coefficients is based on a corresponding coefficient of theoriginal set of coefficients and a corresponding polynomial; and codefor determining the induced high order aberration based on thetransformed set of coefficients.
 19. The computer program productaccording to claim 18, further comprising code for determining atreatment based on the induced high order aberration.
 20. The computerprogram product according to claim 18, further comprising code fordisplaying the transformed set of coefficients for the basis function.